| L(s) = 1 | − 1.19·3-s − 1.19·7-s − 1.56·9-s + 5.86·11-s − 0.364·13-s + 1.19·17-s + 19-s + 1.43·21-s + 8.23·23-s + 5.46·27-s − 7.86·29-s − 7.30·31-s − 7.03·33-s + 7.13·37-s + 0.436·39-s + 2.43·41-s + 7.39·43-s + 13.7·47-s − 5.56·49-s − 1.43·51-s − 7.39·53-s − 1.19·57-s − 12.8·59-s − 1.30·61-s + 1.87·63-s − 11.9·67-s − 9.86·69-s + ⋯ |
| L(s) = 1 | − 0.692·3-s − 0.453·7-s − 0.521·9-s + 1.76·11-s − 0.101·13-s + 0.290·17-s + 0.229·19-s + 0.313·21-s + 1.71·23-s + 1.05·27-s − 1.46·29-s − 1.31·31-s − 1.22·33-s + 1.17·37-s + 0.0699·39-s + 0.380·41-s + 1.12·43-s + 1.99·47-s − 0.794·49-s − 0.201·51-s − 1.01·53-s − 0.158·57-s − 1.67·59-s − 0.166·61-s + 0.236·63-s − 1.45·67-s − 1.18·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.469400039\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.469400039\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 1.19T + 3T^{2} \) |
| 7 | \( 1 + 1.19T + 7T^{2} \) |
| 11 | \( 1 - 5.86T + 11T^{2} \) |
| 13 | \( 1 + 0.364T + 13T^{2} \) |
| 17 | \( 1 - 1.19T + 17T^{2} \) |
| 23 | \( 1 - 8.23T + 23T^{2} \) |
| 29 | \( 1 + 7.86T + 29T^{2} \) |
| 31 | \( 1 + 7.30T + 31T^{2} \) |
| 37 | \( 1 - 7.13T + 37T^{2} \) |
| 41 | \( 1 - 2.43T + 41T^{2} \) |
| 43 | \( 1 - 7.39T + 43T^{2} \) |
| 47 | \( 1 - 13.7T + 47T^{2} \) |
| 53 | \( 1 + 7.39T + 53T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 + 1.30T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 + 2.12T + 71T^{2} \) |
| 73 | \( 1 - 2.50T + 73T^{2} \) |
| 79 | \( 1 - 7.74T + 79T^{2} \) |
| 83 | \( 1 - 3.02T + 83T^{2} \) |
| 89 | \( 1 + 5.68T + 89T^{2} \) |
| 97 | \( 1 - 1.09T + 97T^{2} \) |
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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
−7.59787608423089678889048046062, −7.20279465082218790374115710318, −6.26219957888206299547120104688, −5.98152244628634054109554951483, −5.16857182462666947910263707181, −4.32798573214418665451654452009, −3.55424844461036963261181383574, −2.82656049234235038194951431632, −1.56790185805339350452333605946, −0.66051616729244648808885548679,
0.66051616729244648808885548679, 1.56790185805339350452333605946, 2.82656049234235038194951431632, 3.55424844461036963261181383574, 4.32798573214418665451654452009, 5.16857182462666947910263707181, 5.98152244628634054109554951483, 6.26219957888206299547120104688, 7.20279465082218790374115710318, 7.59787608423089678889048046062
