| L(s) = 1 | − 2-s + 4-s + 0.753·5-s − 4.69·7-s − 8-s − 0.753·10-s + 5.51·11-s − 4.13·13-s + 4.69·14-s + 16-s − 4.34·17-s + 5.65·19-s + 0.753·20-s − 5.51·22-s + 5.69·23-s − 4.43·25-s + 4.13·26-s − 4.69·28-s + 0.417·29-s − 1.55·31-s − 32-s + 4.34·34-s − 3.53·35-s − 8.89·37-s − 5.65·38-s − 0.753·40-s + 9.36·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.336·5-s − 1.77·7-s − 0.353·8-s − 0.238·10-s + 1.66·11-s − 1.14·13-s + 1.25·14-s + 0.250·16-s − 1.05·17-s + 1.29·19-s + 0.168·20-s − 1.17·22-s + 1.18·23-s − 0.886·25-s + 0.811·26-s − 0.886·28-s + 0.0776·29-s − 0.279·31-s − 0.176·32-s + 0.745·34-s − 0.597·35-s − 1.46·37-s − 0.917·38-s − 0.119·40-s + 1.46·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9910089774\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9910089774\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 223 | \( 1 + T \) |
| good | 5 | \( 1 - 0.753T + 5T^{2} \) |
| 7 | \( 1 + 4.69T + 7T^{2} \) |
| 11 | \( 1 - 5.51T + 11T^{2} \) |
| 13 | \( 1 + 4.13T + 13T^{2} \) |
| 17 | \( 1 + 4.34T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 - 5.69T + 23T^{2} \) |
| 29 | \( 1 - 0.417T + 29T^{2} \) |
| 31 | \( 1 + 1.55T + 31T^{2} \) |
| 37 | \( 1 + 8.89T + 37T^{2} \) |
| 41 | \( 1 - 9.36T + 41T^{2} \) |
| 43 | \( 1 - 1.04T + 43T^{2} \) |
| 47 | \( 1 + 2.57T + 47T^{2} \) |
| 53 | \( 1 - 8.32T + 53T^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 61 | \( 1 + 15.5T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 - 1.53T + 73T^{2} \) |
| 79 | \( 1 - 6.46T + 79T^{2} \) |
| 83 | \( 1 + 2.93T + 83T^{2} \) |
| 89 | \( 1 - 4.59T + 89T^{2} \) |
| 97 | \( 1 - 3.37T + 97T^{2} \) |
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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
−8.894387225764684876445903169299, −7.43668277378241957752389262892, −7.07809406102510344393010337703, −6.38247556382708670843985660491, −5.80899127867548061997912745259, −4.64780150609369799411816868854, −3.58109856421568173675886634642, −2.94217160859557057206835270005, −1.88410877670677171378091907591, −0.62064329126177762457307115345,
0.62064329126177762457307115345, 1.88410877670677171378091907591, 2.94217160859557057206835270005, 3.58109856421568173675886634642, 4.64780150609369799411816868854, 5.80899127867548061997912745259, 6.38247556382708670843985660491, 7.07809406102510344393010337703, 7.43668277378241957752389262892, 8.894387225764684876445903169299
