| L(s) = 1 | − 0.622·5-s + 4.42·7-s − 5.80·11-s + 13-s − 2·17-s − 4.42·19-s + 8.85·23-s − 4.61·25-s − 2·29-s − 7.18·31-s − 2.75·35-s + 0.755·37-s − 3.37·41-s − 7.61·43-s − 1.80·47-s + 12.6·49-s − 4.75·53-s + 3.61·55-s − 11.0·59-s − 8.10·61-s − 0.622·65-s + 8.04·67-s + 7.05·71-s + 7.24·73-s − 25.7·77-s − 12·79-s + 3.05·83-s + ⋯ |
| L(s) = 1 | − 0.278·5-s + 1.67·7-s − 1.75·11-s + 0.277·13-s − 0.485·17-s − 1.01·19-s + 1.84·23-s − 0.922·25-s − 0.371·29-s − 1.29·31-s − 0.465·35-s + 0.124·37-s − 0.527·41-s − 1.16·43-s − 0.263·47-s + 1.80·49-s − 0.653·53-s + 0.487·55-s − 1.43·59-s − 1.03·61-s − 0.0771·65-s + 0.982·67-s + 0.836·71-s + 0.847·73-s − 2.93·77-s − 1.35·79-s + 0.334·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 0.622T + 5T^{2} \) |
| 7 | \( 1 - 4.42T + 7T^{2} \) |
| 11 | \( 1 + 5.80T + 11T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 4.42T + 19T^{2} \) |
| 23 | \( 1 - 8.85T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 7.18T + 31T^{2} \) |
| 37 | \( 1 - 0.755T + 37T^{2} \) |
| 41 | \( 1 + 3.37T + 41T^{2} \) |
| 43 | \( 1 + 7.61T + 43T^{2} \) |
| 47 | \( 1 + 1.80T + 47T^{2} \) |
| 53 | \( 1 + 4.75T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 8.10T + 61T^{2} \) |
| 67 | \( 1 - 8.04T + 67T^{2} \) |
| 71 | \( 1 - 7.05T + 71T^{2} \) |
| 73 | \( 1 - 7.24T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 - 3.05T + 83T^{2} \) |
| 89 | \( 1 - 1.86T + 89T^{2} \) |
| 97 | \( 1 + 0.755T + 97T^{2} \) |
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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
−8.101791113111504924053916294011, −7.61688404820186998899285648138, −6.82492143040815592550082672363, −5.68333500682587525358335729836, −5.01832010177739854468275385703, −4.56184701465775866478694369938, −3.44076801967697816800379304937, −2.36291370244336333494585776893, −1.57938427913110355421802480159, 0,
1.57938427913110355421802480159, 2.36291370244336333494585776893, 3.44076801967697816800379304937, 4.56184701465775866478694369938, 5.01832010177739854468275385703, 5.68333500682587525358335729836, 6.82492143040815592550082672363, 7.61688404820186998899285648138, 8.101791113111504924053916294011
