L(s) = 1 | + 24·25-s + 32·29-s + 32·37-s − 16·53-s − 32·81-s + 48·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 88·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
L(s) = 1 | + 24/5·25-s + 5.94·29-s + 5.26·37-s − 2.19·53-s − 3.55·81-s + 4.36·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 6.76·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(42.88279333\) |
\(L(\frac12)\) |
\(\approx\) |
\(42.88279333\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( ( 1 + 16 T^{4} + p^{4} T^{8} )^{2} \) |
| 5 | \( ( 1 - 12 T^{2} + 78 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 24 T^{2} + 354 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 44 T^{2} + 814 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 48 T^{2} + 1056 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 48 T^{2} + 1136 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 36 T^{2} + 870 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + 28 T^{2} + 1990 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 4 T + p T^{2} )^{8} \) |
| 41 | \( ( 1 - 128 T^{2} + 7296 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 56 T^{2} + 4450 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 140 T^{2} + 9286 T^{4} + 140 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 32 T^{2} + 6640 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 44 T^{2} + 2926 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 156 T^{2} + 13014 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - p T^{2} )^{8} \) |
| 73 | \( ( 1 - 32 T^{2} + 6496 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 236 T^{2} + 25894 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 224 T^{2} + 25072 T^{4} + 224 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 304 T^{2} + 38848 T^{4} - 304 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 288 T^{2} + 38304 T^{4} - 288 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−3.48089091886406156716039238064, −3.27331356086466243900511380116, −3.19651304125086668074323111146, −3.14137315863814565354305499500, −3.09622853297910517904012057865, −3.01425763608413461662968448422, −3.01154064161068046486246859782, −2.66453851019187254175272130088, −2.65971762806391538109163221990, −2.65492298071830704570442344794, −2.46150210732057981425940235589, −2.39767946479423931207129493325, −2.38365577607367737412148308850, −1.94357067917664449564389292490, −1.75844855584303169486035714766, −1.58555692437347888327444806492, −1.57853878794230075198902369010, −1.43439989388875242403955704948, −1.23252509175208163959609707858, −1.00547970102265980263832673278, −0.901712979031246792847353060272, −0.69655544460480092722047955769, −0.65291030501069589984515797984, −0.60740410860547546651219720945, −0.39566879361475298954293762748,
0.39566879361475298954293762748, 0.60740410860547546651219720945, 0.65291030501069589984515797984, 0.69655544460480092722047955769, 0.901712979031246792847353060272, 1.00547970102265980263832673278, 1.23252509175208163959609707858, 1.43439989388875242403955704948, 1.57853878794230075198902369010, 1.58555692437347888327444806492, 1.75844855584303169486035714766, 1.94357067917664449564389292490, 2.38365577607367737412148308850, 2.39767946479423931207129493325, 2.46150210732057981425940235589, 2.65492298071830704570442344794, 2.65971762806391538109163221990, 2.66453851019187254175272130088, 3.01154064161068046486246859782, 3.01425763608413461662968448422, 3.09622853297910517904012057865, 3.14137315863814565354305499500, 3.19651304125086668074323111146, 3.27331356086466243900511380116, 3.48089091886406156716039238064
Plot not available for L-functions of degree greater than 10.