| L(s) = 1 | − i·2-s − 4-s − 1.42·5-s + (2.62 + 0.317i)7-s + i·8-s + 1.42i·10-s − 1.96i·11-s + 5.57i·13-s + (0.317 − 2.62i)14-s + 16-s − 1.24·17-s − i·19-s + 1.42·20-s − 1.96·22-s − 0.582i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 0.637·5-s + (0.992 + 0.120i)7-s + 0.353i·8-s + 0.450i·10-s − 0.592i·11-s + 1.54i·13-s + (0.0849 − 0.701i)14-s + 0.250·16-s − 0.302·17-s − 0.229i·19-s + 0.318·20-s − 0.418·22-s − 0.121i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.741i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.671 - 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.117942067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.117942067\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.62 - 0.317i)T \) |
| 19 | \( 1 + iT \) |
| good | 5 | \( 1 + 1.42T + 5T^{2} \) |
| 11 | \( 1 + 1.96iT - 11T^{2} \) |
| 13 | \( 1 - 5.57iT - 13T^{2} \) |
| 17 | \( 1 + 1.24T + 17T^{2} \) |
| 23 | \( 1 + 0.582iT - 23T^{2} \) |
| 29 | \( 1 - 0.497iT - 29T^{2} \) |
| 31 | \( 1 - 6.46iT - 31T^{2} \) |
| 37 | \( 1 - 4.52T + 37T^{2} \) |
| 41 | \( 1 - 1.13T + 41T^{2} \) |
| 43 | \( 1 + 9.71T + 43T^{2} \) |
| 47 | \( 1 + 3.38T + 47T^{2} \) |
| 53 | \( 1 - 9.99iT - 53T^{2} \) |
| 59 | \( 1 + 5.26T + 59T^{2} \) |
| 61 | \( 1 - 11.8iT - 61T^{2} \) |
| 67 | \( 1 + 2.22T + 67T^{2} \) |
| 71 | \( 1 - 4.00iT - 71T^{2} \) |
| 73 | \( 1 + 1.22iT - 73T^{2} \) |
| 79 | \( 1 - 3.74T + 79T^{2} \) |
| 83 | \( 1 + 6.23T + 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 - 4.51iT - 97T^{2} \) |
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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
−8.943254143505121396872242316011, −8.508722668490018403293214250084, −7.68325392640524942803970121416, −6.84801994977604326125793231057, −5.84658466906994546759220925316, −4.75700315378559874742793042925, −4.29870143430070235618638076242, −3.34079423489215350975620460296, −2.20581805975208287915756338044, −1.25526494516584561676765331975,
0.40698821072471354150504664155, 1.89923982881602687679922611525, 3.28145826547777769406976533711, 4.21829087256910148156580673313, 4.95110805128066996186042028687, 5.67661520438924793556313162622, 6.61072857722977350313699678716, 7.66528910084272749173701514800, 7.87194051210311454168288078972, 8.506881171695572060195704853492
