| L(s) = 1 | + (2.23 + 0.119i)5-s − 0.415i·7-s − 11-s − 4i·13-s + 6.51i·17-s − 5.20·19-s + 8.54i·23-s + (4.97 + 0.535i)25-s + 0.895·29-s − 6.73·31-s + (0.0498 − 0.928i)35-s + 8.96i·37-s − 10.0·41-s + 4.78i·43-s − 5.61i·47-s + ⋯ |
| L(s) = 1 | + (0.998 + 0.0536i)5-s − 0.157i·7-s − 0.301·11-s − 1.10i·13-s + 1.58i·17-s − 1.19·19-s + 1.78i·23-s + (0.994 + 0.107i)25-s + 0.166·29-s − 1.21·31-s + (0.00843 − 0.156i)35-s + 1.47i·37-s − 1.57·41-s + 0.730i·43-s − 0.819i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0536 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0536 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.634794409\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.634794409\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.23 - 0.119i)T \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + 0.415iT - 7T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 6.51iT - 17T^{2} \) |
| 19 | \( 1 + 5.20T + 19T^{2} \) |
| 23 | \( 1 - 8.54iT - 23T^{2} \) |
| 29 | \( 1 - 0.895T + 29T^{2} \) |
| 31 | \( 1 + 6.73T + 31T^{2} \) |
| 37 | \( 1 - 8.96iT - 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 4.78iT - 43T^{2} \) |
| 47 | \( 1 + 5.61iT - 47T^{2} \) |
| 53 | \( 1 - 10.0iT - 53T^{2} \) |
| 59 | \( 1 + 1.63T + 59T^{2} \) |
| 61 | \( 1 - 7.10T + 61T^{2} \) |
| 67 | \( 1 - 10.6iT - 67T^{2} \) |
| 71 | \( 1 - 6.19T + 71T^{2} \) |
| 73 | \( 1 + 3.16iT - 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 16.2iT - 83T^{2} \) |
| 89 | \( 1 - 9.56T + 89T^{2} \) |
| 97 | \( 1 - 0.591iT - 97T^{2} \) |
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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
−8.522349203719269741357099414061, −8.063027510291093673192468504113, −7.10173894331921877230091625946, −6.33154047543563406544675224907, −5.65154420082021745540229587509, −5.12750095939635252067925756847, −3.95623531235698856496547328772, −3.18307526867450662254830684866, −2.11619627420082472966996040053, −1.30453562676828394749474654898,
0.44695352635021610572848223366, 2.05063792793275134268592518253, 2.37248300343948635690673821521, 3.64350898320535029479252731615, 4.71066380048728573294697544424, 5.17399976338723382185555381468, 6.19302297176091863006783623525, 6.71909097768333334821920630110, 7.38928328018178246893432168679, 8.562430904056195594316314626142
