| L(s) = 1 | + 2-s + 3-s + 4-s − 0.612·5-s + 6-s − 3.48·7-s + 8-s + 9-s − 0.612·10-s + 5.85·11-s + 12-s + 13-s − 3.48·14-s − 0.612·15-s + 16-s − 4.41·17-s + 18-s − 6.11·19-s − 0.612·20-s − 3.48·21-s + 5.85·22-s + 5.11·23-s + 24-s − 4.62·25-s + 26-s + 27-s − 3.48·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.273·5-s + 0.408·6-s − 1.31·7-s + 0.353·8-s + 0.333·9-s − 0.193·10-s + 1.76·11-s + 0.288·12-s + 0.277·13-s − 0.931·14-s − 0.158·15-s + 0.250·16-s − 1.07·17-s + 0.235·18-s − 1.40·19-s − 0.136·20-s − 0.760·21-s + 1.24·22-s + 1.06·23-s + 0.204·24-s − 0.924·25-s + 0.196·26-s + 0.192·27-s − 0.658·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 5 | \( 1 + 0.612T + 5T^{2} \) |
| 7 | \( 1 + 3.48T + 7T^{2} \) |
| 11 | \( 1 - 5.85T + 11T^{2} \) |
| 17 | \( 1 + 4.41T + 17T^{2} \) |
| 19 | \( 1 + 6.11T + 19T^{2} \) |
| 23 | \( 1 - 5.11T + 23T^{2} \) |
| 29 | \( 1 + 7.19T + 29T^{2} \) |
| 31 | \( 1 + 4.63T + 31T^{2} \) |
| 37 | \( 1 + 4.90T + 37T^{2} \) |
| 41 | \( 1 + 2.73T + 41T^{2} \) |
| 43 | \( 1 + 6.64T + 43T^{2} \) |
| 47 | \( 1 + 4.75T + 47T^{2} \) |
| 53 | \( 1 - 2.85T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 + 4.74T + 67T^{2} \) |
| 71 | \( 1 + 5.14T + 71T^{2} \) |
| 73 | \( 1 + 2.75T + 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 - 17.5T + 83T^{2} \) |
| 89 | \( 1 - 8.15T + 89T^{2} \) |
| 97 | \( 1 + 16.6T + 97T^{2} \) |
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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
−7.07354299359549164123951611006, −6.79998432936180874519802300968, −6.29641710013496586270938759899, −5.42356564223306916602983850476, −4.33480481752507010113898586945, −3.76581553329608568509169435936, −3.45513896483081056395977119619, −2.34445003433395074056130843594, −1.56638819834020731998080327186, 0,
1.56638819834020731998080327186, 2.34445003433395074056130843594, 3.45513896483081056395977119619, 3.76581553329608568509169435936, 4.33480481752507010113898586945, 5.42356564223306916602983850476, 6.29641710013496586270938759899, 6.79998432936180874519802300968, 7.07354299359549164123951611006
