| L(s) = 1 | + (−1.31 − 1.12i)3-s + 4.10·5-s + 3.88i·7-s + (0.462 + 2.96i)9-s + 4.82i·11-s + i·13-s + (−5.40 − 4.62i)15-s − 4.55i·17-s + 6.90·19-s + (4.37 − 5.11i)21-s − 3.87·23-s + 11.8·25-s + (2.73 − 4.42i)27-s − 5.07·29-s + 1.77i·31-s + ⋯ |
| L(s) = 1 | + (−0.759 − 0.650i)3-s + 1.83·5-s + 1.46i·7-s + (0.154 + 0.988i)9-s + 1.45i·11-s + 0.277i·13-s + (−1.39 − 1.19i)15-s − 1.10i·17-s + 1.58·19-s + (0.954 − 1.11i)21-s − 0.807·23-s + 2.37·25-s + (0.525 − 0.850i)27-s − 0.941·29-s + 0.319i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.431 - 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.431 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.876921690\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.876921690\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.31 + 1.12i)T \) |
| 13 | \( 1 - iT \) |
| good | 5 | \( 1 - 4.10T + 5T^{2} \) |
| 7 | \( 1 - 3.88iT - 7T^{2} \) |
| 11 | \( 1 - 4.82iT - 11T^{2} \) |
| 17 | \( 1 + 4.55iT - 17T^{2} \) |
| 19 | \( 1 - 6.90T + 19T^{2} \) |
| 23 | \( 1 + 3.87T + 23T^{2} \) |
| 29 | \( 1 + 5.07T + 29T^{2} \) |
| 31 | \( 1 - 1.77iT - 31T^{2} \) |
| 37 | \( 1 - 7.22iT - 37T^{2} \) |
| 41 | \( 1 - 0.0163iT - 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 + 3.18T + 47T^{2} \) |
| 53 | \( 1 - 1.84T + 53T^{2} \) |
| 59 | \( 1 - 2.53iT - 59T^{2} \) |
| 61 | \( 1 - 2.09iT - 61T^{2} \) |
| 67 | \( 1 + 1.95T + 67T^{2} \) |
| 71 | \( 1 - 5.39T + 71T^{2} \) |
| 73 | \( 1 - 6.58T + 73T^{2} \) |
| 79 | \( 1 + 7.24iT - 79T^{2} \) |
| 83 | \( 1 - 10.3iT - 83T^{2} \) |
| 89 | \( 1 - 0.624iT - 89T^{2} \) |
| 97 | \( 1 + 3.61T + 97T^{2} \) |
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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
−9.445927262650612269729925199111, −8.321951413703869396742582173937, −7.24916903369803622135726037604, −6.65240518852541665632756943065, −5.88602220921683908423204310645, −5.22056225108098756897703054678, −4.91239083604244533599776711282, −2.86688794567370340652930303720, −2.08414058790782593281495275629, −1.50797433979681047870045741681,
0.69452082701964374911292138213, 1.66588235096906371965729787526, 3.25381148326235381802251964529, 3.88426650482572956254520773162, 5.10363844282581674555116632615, 5.70069910216098271925468050327, 6.20905410374538295077555378352, 7.00346288074576548768508354044, 8.063804150005931420867115595256, 9.057539662774859286468263434867
