L(s) = 1 | − 52.0·2-s + 302.·3-s − 1.38e3·4-s − 2.82e4i·5-s − 1.57e4·6-s − 5.21e4i·7-s + 2.85e5·8-s − 4.40e5·9-s + 1.47e6i·10-s − 1.24e6i·11-s − 4.18e5·12-s + 6.44e6·13-s + 2.71e6i·14-s − 8.54e6i·15-s − 9.18e6·16-s − 3.13e7i·17-s + ⋯ |
L(s) = 1 | − 0.813·2-s + 0.414·3-s − 0.338·4-s − 1.80i·5-s − 0.337·6-s − 0.442i·7-s + 1.08·8-s − 0.828·9-s + 1.47i·10-s − 0.702i·11-s − 0.140·12-s + 1.33·13-s + 0.360i·14-s − 0.750i·15-s − 0.547·16-s − 1.30i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 - 0.275i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{13}{2})\) |
\(\approx\) |
\(0.0832523 + 0.591659i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0832523 + 0.591659i\) |
\(L(7)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 + (1.42e8 + 4.08e7i)T \) |
good | 2 | \( 1 + 52.0T + 4.09e3T^{2} \) |
| 3 | \( 1 - 302.T + 5.31e5T^{2} \) |
| 5 | \( 1 + 2.82e4iT - 2.44e8T^{2} \) |
| 7 | \( 1 + 5.21e4iT - 1.38e10T^{2} \) |
| 11 | \( 1 + 1.24e6iT - 3.13e12T^{2} \) |
| 13 | \( 1 - 6.44e6T + 2.32e13T^{2} \) |
| 17 | \( 1 + 3.13e7iT - 5.82e14T^{2} \) |
| 19 | \( 1 - 5.58e7iT - 2.21e15T^{2} \) |
| 29 | \( 1 + 4.81e8T + 3.53e17T^{2} \) |
| 31 | \( 1 - 6.23e8T + 7.87e17T^{2} \) |
| 37 | \( 1 - 2.46e9iT - 6.58e18T^{2} \) |
| 41 | \( 1 + 1.40e9T + 2.25e19T^{2} \) |
| 43 | \( 1 + 3.48e8iT - 3.99e19T^{2} \) |
| 47 | \( 1 + 1.11e10T + 1.16e20T^{2} \) |
| 53 | \( 1 + 2.88e10iT - 4.91e20T^{2} \) |
| 59 | \( 1 - 5.12e10T + 1.77e21T^{2} \) |
| 61 | \( 1 - 3.55e10iT - 2.65e21T^{2} \) |
| 67 | \( 1 - 1.31e11iT - 8.18e21T^{2} \) |
| 71 | \( 1 + 1.14e11T + 1.64e22T^{2} \) |
| 73 | \( 1 - 3.09e8T + 2.29e22T^{2} \) |
| 79 | \( 1 - 1.20e10iT - 5.90e22T^{2} \) |
| 83 | \( 1 - 5.91e11iT - 1.06e23T^{2} \) |
| 89 | \( 1 - 2.28e11iT - 2.46e23T^{2} \) |
| 97 | \( 1 + 5.88e11iT - 6.93e23T^{2} \) |
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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
−13.92848314593100647547318135998, −13.28843316341824506744808304359, −11.60733461250515114240015805079, −9.797903528286738811612618387917, −8.609800259091772709117577529005, −8.182056767703352734039874076283, −5.50505928377567653850468758354, −3.96191424100504528139171082043, −1.31801609756243297996660391261, −0.29204657240153389863391743724,
2.11026369818233135201009719764, 3.65270651050422047811201520421, 6.16900791161373660311433960774, 7.71222692901072897668837136574, 8.942254507819751938757283745109, 10.33211951300992621882332443945, 11.29332615393601069630251941483, 13.45857106263178633595893648551, 14.50534150143658622999505808999, 15.49767146205874588767792615402