| L(s) = 1 | − 0.806·3-s − 4.15·7-s − 2.35·9-s + 4.15·11-s + 2.96·13-s + 0.156·17-s + 0.806·19-s + 3.35·21-s + 0.806·23-s + 4.31·27-s + 29-s − 0.806·31-s − 3.35·33-s − 8.15·37-s − 2.38·39-s + 3.73·41-s + 10.7·43-s − 1.58·47-s + 10.2·49-s − 0.126·51-s − 12.0·53-s − 0.649·57-s − 1.73·59-s − 14.4·61-s + 9.76·63-s + 0.806·67-s − 0.649·69-s + ⋯ |
| L(s) = 1 | − 0.465·3-s − 1.57·7-s − 0.783·9-s + 1.25·11-s + 0.821·13-s + 0.0379·17-s + 0.184·19-s + 0.731·21-s + 0.168·23-s + 0.829·27-s + 0.185·29-s − 0.144·31-s − 0.583·33-s − 1.34·37-s − 0.382·39-s + 0.583·41-s + 1.63·43-s − 0.230·47-s + 1.46·49-s − 0.0176·51-s − 1.65·53-s − 0.0860·57-s − 0.226·59-s − 1.85·61-s + 1.23·63-s + 0.0984·67-s − 0.0782·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 0.806T + 3T^{2} \) |
| 7 | \( 1 + 4.15T + 7T^{2} \) |
| 11 | \( 1 - 4.15T + 11T^{2} \) |
| 13 | \( 1 - 2.96T + 13T^{2} \) |
| 17 | \( 1 - 0.156T + 17T^{2} \) |
| 19 | \( 1 - 0.806T + 19T^{2} \) |
| 23 | \( 1 - 0.806T + 23T^{2} \) |
| 31 | \( 1 + 0.806T + 31T^{2} \) |
| 37 | \( 1 + 8.15T + 37T^{2} \) |
| 41 | \( 1 - 3.73T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 + 1.58T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 + 1.73T + 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 - 0.806T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 + 0.806T + 73T^{2} \) |
| 79 | \( 1 - 0.468T + 79T^{2} \) |
| 83 | \( 1 + 6.08T + 83T^{2} \) |
| 89 | \( 1 + 13.0T + 89T^{2} \) |
| 97 | \( 1 - 4.28T + 97T^{2} \) |
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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
−8.648206344278168980929432647863, −7.49174552255291413024716506086, −6.59235770233858735069120404144, −6.20275383608639098309924767078, −5.57489758619832174750698029945, −4.35394643716318675240542944555, −3.48773046956444016508834407222, −2.85891834459956969505739838172, −1.30114168867141345690225668634, 0,
1.30114168867141345690225668634, 2.85891834459956969505739838172, 3.48773046956444016508834407222, 4.35394643716318675240542944555, 5.57489758619832174750698029945, 6.20275383608639098309924767078, 6.59235770233858735069120404144, 7.49174552255291413024716506086, 8.648206344278168980929432647863
