| L(s) = 1 | − 0.788i·3-s + 0.604·5-s + (−2.57 − 0.620i)7-s + 2.37·9-s − 5.13·11-s + 13-s − 0.476i·15-s − 5.35i·17-s + 5.97i·19-s + (−0.489 + 2.02i)21-s + 1.05i·23-s − 4.63·25-s − 4.23i·27-s + 7.55i·29-s − 2.56·31-s + ⋯ |
| L(s) = 1 | − 0.455i·3-s + 0.270·5-s + (−0.972 − 0.234i)7-s + 0.792·9-s − 1.54·11-s + 0.277·13-s − 0.122i·15-s − 1.29i·17-s + 1.37i·19-s + (−0.106 + 0.442i)21-s + 0.220i·23-s − 0.927·25-s − 0.815i·27-s + 1.40i·29-s − 0.461·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.153 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.153 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8706182248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8706182248\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (2.57 + 0.620i)T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 0.788iT - 3T^{2} \) |
| 5 | \( 1 - 0.604T + 5T^{2} \) |
| 11 | \( 1 + 5.13T + 11T^{2} \) |
| 17 | \( 1 + 5.35iT - 17T^{2} \) |
| 19 | \( 1 - 5.97iT - 19T^{2} \) |
| 23 | \( 1 - 1.05iT - 23T^{2} \) |
| 29 | \( 1 - 7.55iT - 29T^{2} \) |
| 31 | \( 1 + 2.56T + 31T^{2} \) |
| 37 | \( 1 - 7.85iT - 37T^{2} \) |
| 41 | \( 1 + 1.72iT - 41T^{2} \) |
| 43 | \( 1 - 6.68T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 - 4.72iT - 53T^{2} \) |
| 59 | \( 1 + 1.93iT - 59T^{2} \) |
| 61 | \( 1 + 13.1T + 61T^{2} \) |
| 67 | \( 1 + 7.82T + 67T^{2} \) |
| 71 | \( 1 - 9.20iT - 71T^{2} \) |
| 73 | \( 1 + 16.1iT - 73T^{2} \) |
| 79 | \( 1 - 8.46iT - 79T^{2} \) |
| 83 | \( 1 - 10.2iT - 83T^{2} \) |
| 89 | \( 1 - 12.9iT - 89T^{2} \) |
| 97 | \( 1 - 14.1iT - 97T^{2} \) |
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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
−9.072800322939829664234730094474, −7.85631507203512954118437735123, −7.51944870568342496197520398503, −6.74453154002330682701830552207, −5.88855146183371632190721417845, −5.23330409398952183150247551759, −4.15871795806263573289885329545, −3.20497943971441048826888376214, −2.35292872601530664668311405267, −1.15212358860686676749532246499,
0.28757565613405158373033630388, 2.02993300287324339017936721345, 2.85422070288547088366579208061, 3.90365131823980300126410130549, 4.57326474657849903458575758145, 5.71220468474327394142052860795, 6.05737970546464818918399029238, 7.20593150718467018960030621351, 7.72200558186868877321020685351, 8.791882551034892799325224490666
