| L(s) = 1 | − 0.262·2-s + 3-s − 1.93·4-s + 5-s − 0.262·6-s − 2.78·7-s + 1.03·8-s + 9-s − 0.262·10-s + 1.00·11-s − 1.93·12-s − 4.96·13-s + 0.729·14-s + 15-s + 3.59·16-s + 6.56·17-s − 0.262·18-s − 4.05·19-s − 1.93·20-s − 2.78·21-s − 0.263·22-s − 1.33·23-s + 1.03·24-s + 25-s + 1.30·26-s + 27-s + 5.36·28-s + ⋯ |
| L(s) = 1 | − 0.185·2-s + 0.577·3-s − 0.965·4-s + 0.447·5-s − 0.107·6-s − 1.05·7-s + 0.364·8-s + 0.333·9-s − 0.0829·10-s + 0.303·11-s − 0.557·12-s − 1.37·13-s + 0.194·14-s + 0.258·15-s + 0.897·16-s + 1.59·17-s − 0.0618·18-s − 0.929·19-s − 0.431·20-s − 0.606·21-s − 0.0562·22-s − 0.279·23-s + 0.210·24-s + 0.200·25-s + 0.255·26-s + 0.192·27-s + 1.01·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 + 0.262T + 2T^{2} \) |
| 7 | \( 1 + 2.78T + 7T^{2} \) |
| 11 | \( 1 - 1.00T + 11T^{2} \) |
| 13 | \( 1 + 4.96T + 13T^{2} \) |
| 17 | \( 1 - 6.56T + 17T^{2} \) |
| 19 | \( 1 + 4.05T + 19T^{2} \) |
| 23 | \( 1 + 1.33T + 23T^{2} \) |
| 29 | \( 1 + 8.35T + 29T^{2} \) |
| 31 | \( 1 + 7.67T + 31T^{2} \) |
| 37 | \( 1 + 7.04T + 37T^{2} \) |
| 41 | \( 1 - 0.795T + 41T^{2} \) |
| 43 | \( 1 + 7.64T + 43T^{2} \) |
| 47 | \( 1 - 4.83T + 47T^{2} \) |
| 53 | \( 1 + 0.693T + 53T^{2} \) |
| 59 | \( 1 + 8.11T + 59T^{2} \) |
| 61 | \( 1 - 9.14T + 61T^{2} \) |
| 67 | \( 1 + 1.81T + 67T^{2} \) |
| 71 | \( 1 + 7.90T + 71T^{2} \) |
| 73 | \( 1 + 2.81T + 73T^{2} \) |
| 83 | \( 1 + 4.85T + 83T^{2} \) |
| 89 | \( 1 + 6.48T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
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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
−9.395664555850128198856564773518, −8.789325486387281910077925708242, −7.73712779688355789300069909427, −7.07935275709300920691116813989, −5.86955904155512848797706586108, −5.09841807633261990905587213416, −3.91590995756127308445025644850, −3.17971382538798642002584049925, −1.80832671651639209720123290508, 0,
1.80832671651639209720123290508, 3.17971382538798642002584049925, 3.91590995756127308445025644850, 5.09841807633261990905587213416, 5.86955904155512848797706586108, 7.07935275709300920691116813989, 7.73712779688355789300069909427, 8.789325486387281910077925708242, 9.395664555850128198856564773518
