L(s) = 1 | + 8·2-s + 82·3-s + 64·4-s + 656·6-s + 343·7-s + 512·8-s + 4.53e3·9-s + 2.40e3·11-s + 5.24e3·12-s − 7.11e3·13-s + 2.74e3·14-s + 4.09e3·16-s − 2.48e3·17-s + 3.62e4·18-s + 3.64e4·19-s + 2.81e4·21-s + 1.92e4·22-s + 1.28e4·23-s + 4.19e4·24-s − 5.69e4·26-s + 1.92e5·27-s + 2.19e4·28-s − 8.80e4·29-s + 2.82e5·31-s + 3.27e4·32-s + 1.97e5·33-s − 1.98e4·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.75·3-s + 1/2·4-s + 1.23·6-s + 0.377·7-s + 0.353·8-s + 2.07·9-s + 0.545·11-s + 0.876·12-s − 0.898·13-s + 0.267·14-s + 1/4·16-s − 0.122·17-s + 1.46·18-s + 1.22·19-s + 0.662·21-s + 0.385·22-s + 0.220·23-s + 0.619·24-s − 0.635·26-s + 1.88·27-s + 0.188·28-s − 0.670·29-s + 1.70·31-s + 0.176·32-s + 0.956·33-s − 0.0867·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(8.102557407\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.102557407\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - p^{3} T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - p^{3} T \) |
good | 3 | \( 1 - 82 T + p^{7} T^{2} \) |
| 11 | \( 1 - 2408 T + p^{7} T^{2} \) |
| 13 | \( 1 + 7116 T + p^{7} T^{2} \) |
| 17 | \( 1 + 2486 T + p^{7} T^{2} \) |
| 19 | \( 1 - 36482 T + p^{7} T^{2} \) |
| 23 | \( 1 - 560 p T + p^{7} T^{2} \) |
| 29 | \( 1 + 88094 T + p^{7} T^{2} \) |
| 31 | \( 1 - 282636 T + p^{7} T^{2} \) |
| 37 | \( 1 - 214534 T + p^{7} T^{2} \) |
| 41 | \( 1 + 140874 T + p^{7} T^{2} \) |
| 43 | \( 1 + 848 p T + p^{7} T^{2} \) |
| 47 | \( 1 + 716868 T + p^{7} T^{2} \) |
| 53 | \( 1 - 56946 T + p^{7} T^{2} \) |
| 59 | \( 1 + 2149862 T + p^{7} T^{2} \) |
| 61 | \( 1 - 3084360 T + p^{7} T^{2} \) |
| 67 | \( 1 - 3034364 T + p^{7} T^{2} \) |
| 71 | \( 1 + 106624 T + p^{7} T^{2} \) |
| 73 | \( 1 + 988930 T + p^{7} T^{2} \) |
| 79 | \( 1 - 3415896 T + p^{7} T^{2} \) |
| 83 | \( 1 - 15142 T + p^{7} T^{2} \) |
| 89 | \( 1 - 174810 T + p^{7} T^{2} \) |
| 97 | \( 1 + 13506790 T + p^{7} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.982239273512018540142887522468, −9.393123106566862568910299299278, −8.283639899253426181926521537615, −7.58032331401675064331509092974, −6.67900794450008966774296401055, −5.10117934638576850928606136724, −4.13285164499215782834597883342, −3.15415952750196815640144555132, −2.33888454685084093184376309542, −1.24322877861927529392086457022,
1.24322877861927529392086457022, 2.33888454685084093184376309542, 3.15415952750196815640144555132, 4.13285164499215782834597883342, 5.10117934638576850928606136724, 6.67900794450008966774296401055, 7.58032331401675064331509092974, 8.283639899253426181926521537615, 9.393123106566862568910299299278, 9.982239273512018540142887522468