| L(s) = 1 | + 2.29·2-s + 3.28·4-s + 3.80·5-s + 2.12·7-s + 2.96·8-s + 8.73·10-s − 4.17·11-s + 1.38·13-s + 4.88·14-s + 0.232·16-s − 4.82·17-s + 4.03·19-s + 12.4·20-s − 9.60·22-s − 3.89·23-s + 9.44·25-s + 3.19·26-s + 6.97·28-s + 8.74·29-s + 8.80·31-s − 5.38·32-s − 11.0·34-s + 8.06·35-s + 5.73·37-s + 9.28·38-s + 11.2·40-s − 41-s + ⋯ |
| L(s) = 1 | + 1.62·2-s + 1.64·4-s + 1.69·5-s + 0.802·7-s + 1.04·8-s + 2.76·10-s − 1.25·11-s + 0.385·13-s + 1.30·14-s + 0.0581·16-s − 1.16·17-s + 0.926·19-s + 2.79·20-s − 2.04·22-s − 0.812·23-s + 1.88·25-s + 0.626·26-s + 1.31·28-s + 1.62·29-s + 1.58·31-s − 0.952·32-s − 1.90·34-s + 1.36·35-s + 0.942·37-s + 1.50·38-s + 1.77·40-s − 0.156·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3321 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3321 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.879594948\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.879594948\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - 2.29T + 2T^{2} \) |
| 5 | \( 1 - 3.80T + 5T^{2} \) |
| 7 | \( 1 - 2.12T + 7T^{2} \) |
| 11 | \( 1 + 4.17T + 11T^{2} \) |
| 13 | \( 1 - 1.38T + 13T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 19 | \( 1 - 4.03T + 19T^{2} \) |
| 23 | \( 1 + 3.89T + 23T^{2} \) |
| 29 | \( 1 - 8.74T + 29T^{2} \) |
| 31 | \( 1 - 8.80T + 31T^{2} \) |
| 37 | \( 1 - 5.73T + 37T^{2} \) |
| 43 | \( 1 - 6.23T + 43T^{2} \) |
| 47 | \( 1 - 0.698T + 47T^{2} \) |
| 53 | \( 1 + 8.97T + 53T^{2} \) |
| 59 | \( 1 - 4.25T + 59T^{2} \) |
| 61 | \( 1 - 5.62T + 61T^{2} \) |
| 67 | \( 1 + 8.25T + 67T^{2} \) |
| 71 | \( 1 + 3.11T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 2.82T + 79T^{2} \) |
| 83 | \( 1 - 6.23T + 83T^{2} \) |
| 89 | \( 1 + 0.463T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−8.541277852688932143476264989401, −7.75727980208236242791351675983, −6.61994518410291241436892099224, −6.17581094900640527302369378103, −5.44867320568098220028629442893, −4.88958681090127404480497633029, −4.28476085804734076002649351906, −2.77445789939449148420502351437, −2.52684213886308804457869355416, −1.43138066176782043558521166995,
1.43138066176782043558521166995, 2.52684213886308804457869355416, 2.77445789939449148420502351437, 4.28476085804734076002649351906, 4.88958681090127404480497633029, 5.44867320568098220028629442893, 6.17581094900640527302369378103, 6.61994518410291241436892099224, 7.75727980208236242791351675983, 8.541277852688932143476264989401
