| L(s) = 1 | − 2-s + 4-s − 1.77·5-s + 1.25·7-s − 8-s + 1.77·10-s − 2.17·11-s + 0.350·13-s − 1.25·14-s + 16-s − 0.495·17-s + 2.51·19-s − 1.77·20-s + 2.17·22-s − 1.85·25-s − 0.350·26-s + 1.25·28-s − 0.774·29-s + 4.19·31-s − 32-s + 0.495·34-s − 2.22·35-s − 2.77·37-s − 2.51·38-s + 1.77·40-s − 8.15·41-s + 10.0·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.792·5-s + 0.474·7-s − 0.353·8-s + 0.560·10-s − 0.655·11-s + 0.0971·13-s − 0.335·14-s + 0.250·16-s − 0.120·17-s + 0.575·19-s − 0.396·20-s + 0.463·22-s − 0.371·25-s − 0.0687·26-s + 0.237·28-s − 0.143·29-s + 0.754·31-s − 0.176·32-s + 0.0849·34-s − 0.376·35-s − 0.455·37-s − 0.407·38-s + 0.280·40-s − 1.27·41-s + 1.52·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 1.77T + 5T^{2} \) |
| 7 | \( 1 - 1.25T + 7T^{2} \) |
| 11 | \( 1 + 2.17T + 11T^{2} \) |
| 13 | \( 1 - 0.350T + 13T^{2} \) |
| 17 | \( 1 + 0.495T + 17T^{2} \) |
| 19 | \( 1 - 2.51T + 19T^{2} \) |
| 29 | \( 1 + 0.774T + 29T^{2} \) |
| 31 | \( 1 - 4.19T + 31T^{2} \) |
| 37 | \( 1 + 2.77T + 37T^{2} \) |
| 41 | \( 1 + 8.15T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 - 7.77T + 47T^{2} \) |
| 53 | \( 1 - 5.99T + 53T^{2} \) |
| 59 | \( 1 + 3.35T + 59T^{2} \) |
| 61 | \( 1 + 8.58T + 61T^{2} \) |
| 67 | \( 1 - 4.58T + 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 - 4.91T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 + 0.674T + 97T^{2} \) |
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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
−7.54782005041684982668747183952, −6.93383145629221165288296308183, −6.05005284269777856896167846049, −5.32778247840530860447303523333, −4.56151150072777038481006422960, −3.75904982612414387221376605267, −2.94517064562137940127330796954, −2.09081243277305316383276992632, −1.06287565116037406026740719785, 0,
1.06287565116037406026740719785, 2.09081243277305316383276992632, 2.94517064562137940127330796954, 3.75904982612414387221376605267, 4.56151150072777038481006422960, 5.32778247840530860447303523333, 6.05005284269777856896167846049, 6.93383145629221165288296308183, 7.54782005041684982668747183952
