L(s) = 1 | + 2-s − 3.36·3-s + 4-s + 2.69·5-s − 3.36·6-s − 4.35·7-s + 8-s + 8.30·9-s + 2.69·10-s − 4.44·11-s − 3.36·12-s − 2.85·13-s − 4.35·14-s − 9.05·15-s + 16-s − 4.20·17-s + 8.30·18-s − 5.66·19-s + 2.69·20-s + 14.6·21-s − 4.44·22-s − 23-s − 3.36·24-s + 2.25·25-s − 2.85·26-s − 17.8·27-s − 4.35·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.94·3-s + 0.5·4-s + 1.20·5-s − 1.37·6-s − 1.64·7-s + 0.353·8-s + 2.76·9-s + 0.851·10-s − 1.33·11-s − 0.970·12-s − 0.790·13-s − 1.16·14-s − 2.33·15-s + 0.250·16-s − 1.01·17-s + 1.95·18-s − 1.29·19-s + 0.602·20-s + 3.19·21-s − 0.947·22-s − 0.208·23-s − 0.686·24-s + 0.450·25-s − 0.559·26-s − 3.43·27-s − 0.822·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6723554156\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6723554156\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 + T \) |
good | 3 | \( 1 + 3.36T + 3T^{2} \) |
| 5 | \( 1 - 2.69T + 5T^{2} \) |
| 7 | \( 1 + 4.35T + 7T^{2} \) |
| 11 | \( 1 + 4.44T + 11T^{2} \) |
| 13 | \( 1 + 2.85T + 13T^{2} \) |
| 17 | \( 1 + 4.20T + 17T^{2} \) |
| 19 | \( 1 + 5.66T + 19T^{2} \) |
| 29 | \( 1 - 2.22T + 29T^{2} \) |
| 31 | \( 1 + 4.57T + 31T^{2} \) |
| 37 | \( 1 - 1.52T + 37T^{2} \) |
| 41 | \( 1 + 0.472T + 41T^{2} \) |
| 43 | \( 1 - 11.8T + 43T^{2} \) |
| 47 | \( 1 + 2.55T + 47T^{2} \) |
| 53 | \( 1 + 6.65T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 + 8.36T + 61T^{2} \) |
| 67 | \( 1 - 4.53T + 67T^{2} \) |
| 71 | \( 1 + 7.21T + 71T^{2} \) |
| 73 | \( 1 + 9.18T + 73T^{2} \) |
| 79 | \( 1 - 15.7T + 79T^{2} \) |
| 83 | \( 1 - 6.45T + 83T^{2} \) |
| 89 | \( 1 + 4.86T + 89T^{2} \) |
| 97 | \( 1 - 2.32T + 97T^{2} \) |
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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
−7.59881470748800318958520112531, −6.93125596271509567783951334731, −6.23171849275754103762738658226, −6.09418159264406157404947453746, −5.32973789168662850898111563385, −4.74460981087304251345020244034, −3.94153698763417416093873948313, −2.63314442291782372135866773681, −1.99574919658733635314274357459, −0.39726865027670081584152890086,
0.39726865027670081584152890086, 1.99574919658733635314274357459, 2.63314442291782372135866773681, 3.94153698763417416093873948313, 4.74460981087304251345020244034, 5.32973789168662850898111563385, 6.09418159264406157404947453746, 6.23171849275754103762738658226, 6.93125596271509567783951334731, 7.59881470748800318958520112531