| L(s) = 1 | − 2-s − 0.909·3-s + 4-s − 4.17·5-s + 0.909·6-s + 4.37·7-s − 8-s − 2.17·9-s + 4.17·10-s + 11-s − 0.909·12-s + 4.01·13-s − 4.37·14-s + 3.79·15-s + 16-s − 4.33·17-s + 2.17·18-s − 4.17·20-s − 3.98·21-s − 22-s + 1.15·23-s + 0.909·24-s + 12.4·25-s − 4.01·26-s + 4.70·27-s + 4.37·28-s − 7.57·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.525·3-s + 0.5·4-s − 1.86·5-s + 0.371·6-s + 1.65·7-s − 0.353·8-s − 0.724·9-s + 1.32·10-s + 0.301·11-s − 0.262·12-s + 1.11·13-s − 1.16·14-s + 0.980·15-s + 0.250·16-s − 1.05·17-s + 0.511·18-s − 0.933·20-s − 0.868·21-s − 0.213·22-s + 0.240·23-s + 0.185·24-s + 2.48·25-s − 0.786·26-s + 0.905·27-s + 0.826·28-s − 1.40·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 0.909T + 3T^{2} \) |
| 5 | \( 1 + 4.17T + 5T^{2} \) |
| 7 | \( 1 - 4.37T + 7T^{2} \) |
| 13 | \( 1 - 4.01T + 13T^{2} \) |
| 17 | \( 1 + 4.33T + 17T^{2} \) |
| 23 | \( 1 - 1.15T + 23T^{2} \) |
| 29 | \( 1 + 7.57T + 29T^{2} \) |
| 31 | \( 1 - 0.591T + 31T^{2} \) |
| 37 | \( 1 + 4.27T + 37T^{2} \) |
| 41 | \( 1 + 4.60T + 41T^{2} \) |
| 43 | \( 1 - 2.83T + 43T^{2} \) |
| 47 | \( 1 + 4.67T + 47T^{2} \) |
| 53 | \( 1 - 8.70T + 53T^{2} \) |
| 59 | \( 1 - 4.57T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 + 0.442T + 67T^{2} \) |
| 71 | \( 1 - 5.37T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 + 9.76T + 83T^{2} \) |
| 89 | \( 1 - 2.96T + 89T^{2} \) |
| 97 | \( 1 - 5.86T + 97T^{2} \) |
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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
−7.60670081681126920056730028652, −7.06842200766863929008964482884, −6.24282474608147328972108143106, −5.34012511462049778651407312422, −4.63918199277990450413540394875, −3.95624825426937105338118974340, −3.20478181633146081671523994838, −1.96217735757310040073398923913, −0.989230996650914393915736808784, 0,
0.989230996650914393915736808784, 1.96217735757310040073398923913, 3.20478181633146081671523994838, 3.95624825426937105338118974340, 4.63918199277990450413540394875, 5.34012511462049778651407312422, 6.24282474608147328972108143106, 7.06842200766863929008964482884, 7.60670081681126920056730028652
