| L(s) = 1 | − 8·7-s + 8·9-s − 8·17-s − 12·23-s + 16·25-s + 12·31-s + 28·47-s − 64·63-s + 8·73-s + 44·79-s + 16·81-s + 8·89-s + 8·97-s − 32·103-s − 24·113-s + 64·119-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 64·153-s + 157-s + 96·161-s + 163-s + ⋯ |
| L(s) = 1 | − 3.02·7-s + 8/3·9-s − 1.94·17-s − 2.50·23-s + 16/5·25-s + 2.15·31-s + 4.08·47-s − 8.06·63-s + 0.936·73-s + 4.95·79-s + 16/9·81-s + 0.847·89-s + 0.812·97-s − 3.15·103-s − 2.25·113-s + 5.86·119-s + 4·121-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 − 5.17·153-s + 0.0798·157-s + 7.56·161-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.856895737\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.856895737\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( ( 1 + T )^{8} \) |
| good | 3 | \( 1 - 8 T^{2} + 16 p T^{4} - 68 p T^{6} + 694 T^{8} - 68 p^{3} T^{10} + 16 p^{5} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16} \) |
| 5 | \( 1 - 16 T^{2} + 156 T^{4} - 1152 T^{6} + 6534 T^{8} - 1152 p^{2} T^{10} + 156 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} \) |
| 7 | \( ( 1 + 4 T + 24 T^{2} + 66 T^{3} + 246 T^{4} + 66 p T^{5} + 24 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 - 4 p T^{2} + 96 p T^{4} - 17096 T^{6} + 212502 T^{8} - 17096 p^{2} T^{10} + 96 p^{5} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16} \) |
| 13 | \( 1 - 44 T^{2} + 948 T^{4} - 12548 T^{6} + 151254 T^{8} - 12548 p^{2} T^{10} + 948 p^{4} T^{12} - 44 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( 1 - 40 T^{2} + 1276 T^{4} - 34008 T^{6} + 679846 T^{8} - 34008 p^{2} T^{10} + 1276 p^{4} T^{12} - 40 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( ( 1 + 6 T + 80 T^{2} + 344 T^{3} + 2670 T^{4} + 344 p T^{5} + 80 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( 1 - 80 T^{2} + 5052 T^{4} - 207680 T^{6} + 7067910 T^{8} - 207680 p^{2} T^{10} + 5052 p^{4} T^{12} - 80 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( ( 1 - 6 T + 84 T^{2} - 328 T^{3} + 3350 T^{4} - 328 p T^{5} + 84 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 - 232 T^{2} + 25052 T^{4} - 1655656 T^{6} + 73686022 T^{8} - 1655656 p^{2} T^{10} + 25052 p^{4} T^{12} - 232 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( ( 1 + 100 T^{2} - 112 T^{3} + 4790 T^{4} - 112 p T^{5} + 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 - 188 T^{2} + 19716 T^{4} - 1379108 T^{6} + 69340374 T^{8} - 1379108 p^{2} T^{10} + 19716 p^{4} T^{12} - 188 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( ( 1 - 14 T + 224 T^{2} - 1918 T^{3} + 16446 T^{4} - 1918 p T^{5} + 224 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 53 | \( 1 - 248 T^{2} + 31836 T^{4} - 2685704 T^{6} + 164940582 T^{8} - 2685704 p^{2} T^{10} + 31836 p^{4} T^{12} - 248 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( 1 - 332 T^{2} + 53476 T^{4} - 5447828 T^{6} + 382684822 T^{8} - 5447828 p^{2} T^{10} + 53476 p^{4} T^{12} - 332 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( 1 + 16 T^{2} + 1692 T^{4} - 215840 T^{6} + 4429638 T^{8} - 215840 p^{2} T^{10} + 1692 p^{4} T^{12} + 16 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( 1 - 296 T^{2} + 41532 T^{4} - 3814616 T^{6} + 276036006 T^{8} - 3814616 p^{2} T^{10} + 41532 p^{4} T^{12} - 296 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 76 T^{2} + 182 T^{3} + 5318 T^{4} + 182 p T^{5} + 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 4 T - 76 T^{2} + 68 T^{3} + 8950 T^{4} + 68 p T^{5} - 76 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 22 T + 472 T^{2} - 5592 T^{3} + 62310 T^{4} - 5592 p T^{5} + 472 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 220 T^{2} + 36036 T^{4} - 4244676 T^{6} + 399450390 T^{8} - 4244676 p^{2} T^{10} + 36036 p^{4} T^{12} - 220 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 4 T + 268 T^{2} - 592 T^{3} + 31790 T^{4} - 592 p T^{5} + 268 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 4 T + 284 T^{2} - 460 T^{3} + 35110 T^{4} - 460 p T^{5} + 284 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−4.70095501440919061735356867863, −4.42938960723650486618970531838, −4.35116248040500866055690799799, −4.31115566152452635292721562637, −4.18588571844708692624080082022, −4.07674195021167216631488992314, −3.90661006902498133140079678699, −3.89349415364572159818025106369, −3.55074058827552664783312109561, −3.42318645594993070163595255671, −3.23321462374948841724269201696, −3.10456852776623733671916602006, −3.06817252460881717953097474287, −2.80520429469454514504029667688, −2.61522321442525343732650673006, −2.51525000348534827732607379398, −2.38198952632412943191529981581, −1.96091267076993496322054104976, −1.88259257570833605821798174036, −1.84073829733850115754782104755, −1.49363519363242107653055900326, −1.20524252528481370230502414163, −0.73496374368899349875891412508, −0.56031929414764921799188233254, −0.54575225294065320675180440983,
0.54575225294065320675180440983, 0.56031929414764921799188233254, 0.73496374368899349875891412508, 1.20524252528481370230502414163, 1.49363519363242107653055900326, 1.84073829733850115754782104755, 1.88259257570833605821798174036, 1.96091267076993496322054104976, 2.38198952632412943191529981581, 2.51525000348534827732607379398, 2.61522321442525343732650673006, 2.80520429469454514504029667688, 3.06817252460881717953097474287, 3.10456852776623733671916602006, 3.23321462374948841724269201696, 3.42318645594993070163595255671, 3.55074058827552664783312109561, 3.89349415364572159818025106369, 3.90661006902498133140079678699, 4.07674195021167216631488992314, 4.18588571844708692624080082022, 4.31115566152452635292721562637, 4.35116248040500866055690799799, 4.42938960723650486618970531838, 4.70095501440919061735356867863
Plot not available for L-functions of degree greater than 10.
