| L(s) = 1 | + 10·5-s − 4·9-s + 20·23-s + 31·25-s − 8·31-s − 14·37-s + 12·41-s − 6·43-s − 40·45-s + 12·49-s − 6·59-s − 22·61-s + 64·73-s + 10·81-s − 22·83-s + 32·103-s + 34·107-s + 24·113-s + 200·115-s + 49·121-s − 38·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 4.47·5-s − 4/3·9-s + 4.17·23-s + 31/5·25-s − 1.43·31-s − 2.30·37-s + 1.87·41-s − 0.914·43-s − 5.96·45-s + 12/7·49-s − 0.781·59-s − 2.81·61-s + 7.49·73-s + 10/9·81-s − 2.41·83-s + 3.15·103-s + 3.28·107-s + 2.25·113-s + 18.6·115-s + 4.45·121-s − 3.39·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 41^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 41^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(35.93274432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(35.93274432\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T^{2} )^{4} \) |
| 41 | \( 1 - 12 T + 20 T^{2} + 140 T^{3} - 442 T^{4} + 140 p T^{5} + 20 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| good | 5 | \( ( 1 - p T + 22 T^{2} - 61 T^{3} + 32 p T^{4} - 61 p T^{5} + 22 p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} )^{2} \) |
| 7 | \( 1 - 12 T^{2} + 16 T^{4} - 144 T^{6} + 3894 T^{8} - 144 p^{2} T^{10} + 16 p^{4} T^{12} - 12 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( 1 - 49 T^{2} + 1073 T^{4} - 14578 T^{6} + 162606 T^{8} - 14578 p^{2} T^{10} + 1073 p^{4} T^{12} - 49 p^{6} T^{14} + p^{8} T^{16} \) |
| 13 | \( 1 - 59 T^{2} + 1862 T^{4} - 39349 T^{6} + 597958 T^{8} - 39349 p^{2} T^{10} + 1862 p^{4} T^{12} - 59 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( 1 - 76 T^{2} + 2846 T^{4} - 71396 T^{6} + 1362379 T^{8} - 71396 p^{2} T^{10} + 2846 p^{4} T^{12} - 76 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( 1 - 63 T^{2} + 2566 T^{4} - 74329 T^{6} + 1592342 T^{8} - 74329 p^{2} T^{10} + 2566 p^{4} T^{12} - 63 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( ( 1 - 10 T + 4 p T^{2} - 528 T^{3} + 2966 T^{4} - 528 p T^{5} + 4 p^{3} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( 1 - 89 T^{2} + 2797 T^{4} - 9546 T^{6} - 1197310 T^{8} - 9546 p^{2} T^{10} + 2797 p^{4} T^{12} - 89 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( ( 1 + 4 T + 26 T^{2} + 244 T^{3} + 1551 T^{4} + 244 p T^{5} + 26 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 7 T + 125 T^{2} + 562 T^{3} + 6234 T^{4} + 562 p T^{5} + 125 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 3 T + 5 T^{2} + 196 T^{3} + 3150 T^{4} + 196 p T^{5} + 5 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 101 T^{2} + 4517 T^{4} - 250150 T^{6} + 15503614 T^{8} - 250150 p^{2} T^{10} + 4517 p^{4} T^{12} - 101 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( 1 - 372 T^{2} + 62340 T^{4} - 6193308 T^{6} + 401248118 T^{8} - 6193308 p^{2} T^{10} + 62340 p^{4} T^{12} - 372 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( ( 1 + 3 T + 128 T^{2} - 133 T^{3} + 7358 T^{4} - 133 p T^{5} + 128 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 11 T + 131 T^{2} + 558 T^{3} + 6264 T^{4} + 558 p T^{5} + 131 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 - 99 T^{2} + 10702 T^{4} - 597781 T^{6} + 50273666 T^{8} - 597781 p^{2} T^{10} + 10702 p^{4} T^{12} - 99 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( 1 - 484 T^{2} + 107462 T^{4} - 14271332 T^{6} + 1238734771 T^{8} - 14271332 p^{2} T^{10} + 107462 p^{4} T^{12} - 484 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( ( 1 - 32 T + 590 T^{2} - 7368 T^{3} + 71463 T^{4} - 7368 p T^{5} + 590 p^{2} T^{6} - 32 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( 1 - 252 T^{2} + 42356 T^{4} - 4978564 T^{6} + 448301718 T^{8} - 4978564 p^{2} T^{10} + 42356 p^{4} T^{12} - 252 p^{6} T^{14} + p^{8} T^{16} \) |
| 83 | \( ( 1 + 11 T + 300 T^{2} + 2385 T^{3} + 36656 T^{4} + 2385 p T^{5} + 300 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 - 515 T^{2} + 123194 T^{4} - 18352861 T^{6} + 1916026282 T^{8} - 18352861 p^{2} T^{10} + 123194 p^{4} T^{12} - 515 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( 1 - 672 T^{2} + 204320 T^{4} - 36972988 T^{6} + 4377439878 T^{8} - 36972988 p^{2} T^{10} + 204320 p^{4} T^{12} - 672 p^{6} T^{14} + p^{8} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.76983899923220837084452519932, −3.72265103745006881032338370124, −3.44706224171447349294791212979, −3.42004944095392044877130893134, −3.39186211539102985040511223696, −3.27223647722240577700846011444, −3.00190787395448703953627959198, −2.84495574824219940307046293806, −2.78681524950840436050576803511, −2.77441685391413594295679702833, −2.48534988321920390827216898195, −2.37908793791837824646474511091, −2.30026282818392959328523750276, −2.15448403600914245795730529412, −1.91805672166648703867512882680, −1.79001640385807762322110568249, −1.76035496064407236940694164472, −1.73393764049038566672992801628, −1.65273128639942721198650806292, −1.46828344174296232180501763338, −0.932665462420509115971878811351, −0.926565201569196254034921499296, −0.66102074169402319887355837486, −0.57667141643047489914898910989, −0.36702437093821801844431855841,
0.36702437093821801844431855841, 0.57667141643047489914898910989, 0.66102074169402319887355837486, 0.926565201569196254034921499296, 0.932665462420509115971878811351, 1.46828344174296232180501763338, 1.65273128639942721198650806292, 1.73393764049038566672992801628, 1.76035496064407236940694164472, 1.79001640385807762322110568249, 1.91805672166648703867512882680, 2.15448403600914245795730529412, 2.30026282818392959328523750276, 2.37908793791837824646474511091, 2.48534988321920390827216898195, 2.77441685391413594295679702833, 2.78681524950840436050576803511, 2.84495574824219940307046293806, 3.00190787395448703953627959198, 3.27223647722240577700846011444, 3.39186211539102985040511223696, 3.42004944095392044877130893134, 3.44706224171447349294791212979, 3.72265103745006881032338370124, 3.76983899923220837084452519932
Plot not available for L-functions of degree greater than 10.
