| L(s) = 1 | − 3-s − 5-s + 4·7-s + 9-s − 2·13-s + 15-s + 3·17-s − 8·19-s − 4·21-s − 23-s + 25-s − 27-s − 2·29-s + 3·31-s − 4·35-s + 4·37-s + 2·39-s + 6·41-s + 10·43-s − 45-s + 9·47-s + 9·49-s − 3·51-s + 5·53-s + 8·57-s − 6·59-s − 61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.554·13-s + 0.258·15-s + 0.727·17-s − 1.83·19-s − 0.872·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.538·31-s − 0.676·35-s + 0.657·37-s + 0.320·39-s + 0.937·41-s + 1.52·43-s − 0.149·45-s + 1.31·47-s + 9/7·49-s − 0.420·51-s + 0.686·53-s + 1.05·57-s − 0.781·59-s − 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−14.00214228780975, −13.31550714656251, −12.60546442586625, −12.39524765521715, −11.93313376399060, −11.32279208025637, −10.98900282302319, −10.61427310601290, −10.12044058160583, −9.403502526140032, −8.880516641790052, −8.305416913730519, −7.864919015559608, −7.481478806757498, −6.967577746958582, −6.161980522503143, −5.825437073145614, −5.146427404373568, −4.690536612257341, −4.095875363026790, −3.915384144372837, −2.611788930501192, −2.357138214580942, −1.462550457263251, −0.8941718032488336, 0,
0.8941718032488336, 1.462550457263251, 2.357138214580942, 2.611788930501192, 3.915384144372837, 4.095875363026790, 4.690536612257341, 5.146427404373568, 5.825437073145614, 6.161980522503143, 6.967577746958582, 7.481478806757498, 7.864919015559608, 8.305416913730519, 8.880516641790052, 9.403502526140032, 10.12044058160583, 10.61427310601290, 10.98900282302319, 11.32279208025637, 11.93313376399060, 12.39524765521715, 12.60546442586625, 13.31550714656251, 14.00214228780975
