L(s) = 1 | + (−0.869 − 1.49i)3-s − 5-s + (0.807 + 2.51i)7-s + (−1.48 + 2.60i)9-s + 0.541i·11-s + 5.85i·13-s + (0.869 + 1.49i)15-s − 6.22·17-s − 5.74i·19-s + (3.07 − 3.39i)21-s − 6.55i·23-s + 25-s + (5.19 − 0.0312i)27-s − 5.28i·29-s − 8.99i·31-s + ⋯ |
L(s) = 1 | + (−0.501 − 0.865i)3-s − 0.447·5-s + (0.305 + 0.952i)7-s + (−0.496 + 0.868i)9-s + 0.163i·11-s + 1.62i·13-s + (0.224 + 0.386i)15-s − 1.50·17-s − 1.31i·19-s + (0.670 − 0.741i)21-s − 1.36i·23-s + 0.200·25-s + (0.999 − 0.00601i)27-s − 0.981i·29-s − 1.61i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.670 + 0.741i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.670 + 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6005603488\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6005603488\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.869 + 1.49i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (-0.807 - 2.51i)T \) |
good | 11 | \( 1 - 0.541iT - 11T^{2} \) |
| 13 | \( 1 - 5.85iT - 13T^{2} \) |
| 17 | \( 1 + 6.22T + 17T^{2} \) |
| 19 | \( 1 + 5.74iT - 19T^{2} \) |
| 23 | \( 1 + 6.55iT - 23T^{2} \) |
| 29 | \( 1 + 5.28iT - 29T^{2} \) |
| 31 | \( 1 + 8.99iT - 31T^{2} \) |
| 37 | \( 1 - 7.33T + 37T^{2} \) |
| 41 | \( 1 + 5.16T + 41T^{2} \) |
| 43 | \( 1 - 2.48T + 43T^{2} \) |
| 47 | \( 1 + 2.09T + 47T^{2} \) |
| 53 | \( 1 + 9.75iT - 53T^{2} \) |
| 59 | \( 1 + 9.76T + 59T^{2} \) |
| 61 | \( 1 + 0.433iT - 61T^{2} \) |
| 67 | \( 1 - 8.26T + 67T^{2} \) |
| 71 | \( 1 - 5.25iT - 71T^{2} \) |
| 73 | \( 1 + 9.42iT - 73T^{2} \) |
| 79 | \( 1 + 17.3T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 + 7.12T + 89T^{2} \) |
| 97 | \( 1 + 3.24iT - 97T^{2} \) |
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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
−8.870574237033769994875041657082, −8.336563489375595928960353190795, −7.35351656295210379337953963238, −6.56359566920775607412029871765, −6.12540638226655486265294324761, −4.77487558662761527109842869170, −4.38325838372177231417929681914, −2.52180465009433133748637028232, −2.03125627419234307339942445297, −0.26436553751650028920260718514,
1.19246336900877496381146050449, 3.12862759692232532190558641933, 3.77727451869517265094009505243, 4.68976765428443679631412719138, 5.42362126670984516317607742443, 6.33829395408460154174651934872, 7.32255536489549277527101229320, 8.078273065631252452965475819448, 8.863315067840518711620622071808, 9.822882339746547783431492815330