| L(s) = 1 | − 2-s + 2.24·3-s + 4-s − 0.981·5-s − 2.24·6-s + 4.63·7-s − 8-s + 2.05·9-s + 0.981·10-s + 11-s + 2.24·12-s − 2.46·13-s − 4.63·14-s − 2.20·15-s + 16-s − 1.22·17-s − 2.05·18-s − 0.981·20-s + 10.4·21-s − 22-s − 1.01·23-s − 2.24·24-s − 4.03·25-s + 2.46·26-s − 2.12·27-s + 4.63·28-s + 6.45·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.29·3-s + 0.5·4-s − 0.439·5-s − 0.917·6-s + 1.75·7-s − 0.353·8-s + 0.684·9-s + 0.310·10-s + 0.301·11-s + 0.648·12-s − 0.682·13-s − 1.23·14-s − 0.569·15-s + 0.250·16-s − 0.297·17-s − 0.484·18-s − 0.219·20-s + 2.27·21-s − 0.213·22-s − 0.212·23-s − 0.458·24-s − 0.807·25-s + 0.482·26-s − 0.409·27-s + 0.876·28-s + 1.19·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.745654601\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.745654601\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 2.24T + 3T^{2} \) |
| 5 | \( 1 + 0.981T + 5T^{2} \) |
| 7 | \( 1 - 4.63T + 7T^{2} \) |
| 13 | \( 1 + 2.46T + 13T^{2} \) |
| 17 | \( 1 + 1.22T + 17T^{2} \) |
| 23 | \( 1 + 1.01T + 23T^{2} \) |
| 29 | \( 1 - 6.45T + 29T^{2} \) |
| 31 | \( 1 - 4.69T + 31T^{2} \) |
| 37 | \( 1 - 8.14T + 37T^{2} \) |
| 41 | \( 1 - 6.23T + 41T^{2} \) |
| 43 | \( 1 + 6.57T + 43T^{2} \) |
| 47 | \( 1 + 5.52T + 47T^{2} \) |
| 53 | \( 1 - 2.74T + 53T^{2} \) |
| 59 | \( 1 - 9.77T + 59T^{2} \) |
| 61 | \( 1 + 0.443T + 61T^{2} \) |
| 67 | \( 1 + 14.0T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 - 6.16T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 - 5.67T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 - 16.1T + 97T^{2} \) |
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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
−7.82641003831002845709380202289, −7.71211133342156679516118471350, −6.78265948626678373052752607018, −5.83579518763803073832678982371, −4.76486651427018998304330446364, −4.31366599278744108358191553558, −3.34303195168229980503058028827, −2.41006759429599448328850559837, −1.93510102482965518096328655471, −0.875723867130178416161665772029,
0.875723867130178416161665772029, 1.93510102482965518096328655471, 2.41006759429599448328850559837, 3.34303195168229980503058028827, 4.31366599278744108358191553558, 4.76486651427018998304330446364, 5.83579518763803073832678982371, 6.78265948626678373052752607018, 7.71211133342156679516118471350, 7.82641003831002845709380202289
