| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s + 11-s + 12-s − 13-s + 16-s + 17-s + 18-s + 22-s − 3·23-s + 24-s − 26-s + 27-s + 6·29-s − 6·31-s + 32-s + 33-s + 34-s + 36-s − 6·37-s − 39-s − 3·41-s + 2·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s − 0.277·13-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.213·22-s − 0.625·23-s + 0.204·24-s − 0.196·26-s + 0.192·27-s + 1.11·29-s − 1.07·31-s + 0.176·32-s + 0.174·33-s + 0.171·34-s + 1/6·36-s − 0.986·37-s − 0.160·39-s − 0.468·41-s + 0.304·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−12.57733046097827, −12.35643628281748, −12.07848658129890, −11.47441281543810, −10.82417357719966, −10.66712944471420, −9.991673480410106, −9.552630667465885, −9.205797390749827, −8.488727187027511, −8.184003893161509, −7.711963140899422, −7.074752120545610, −6.794737199268891, −6.283823410556519, −5.659372796242071, −5.192983976918750, −4.782852994251065, −4.049786064723045, −3.784556647448255, −3.255987502354175, −2.617042187993248, −2.186210818125148, −1.574073065061765, −0.9449013097640060, 0,
0.9449013097640060, 1.574073065061765, 2.186210818125148, 2.617042187993248, 3.255987502354175, 3.784556647448255, 4.049786064723045, 4.782852994251065, 5.192983976918750, 5.659372796242071, 6.283823410556519, 6.794737199268891, 7.074752120545610, 7.711963140899422, 8.184003893161509, 8.488727187027511, 9.205797390749827, 9.552630667465885, 9.991673480410106, 10.66712944471420, 10.82417357719966, 11.47441281543810, 12.07848658129890, 12.35643628281748, 12.57733046097827
