| L(s) = 1 | − 2.74·2-s − 0.483·4-s + 19.4·5-s − 7.48·7-s + 23.2·8-s − 53.4·10-s + 22.8·11-s + 20.5·14-s − 59.8·16-s − 67.0·17-s − 16.5·19-s − 9.41·20-s − 62.7·22-s + 175.·23-s + 254.·25-s + 3.61·28-s − 291.·29-s − 117.·31-s − 21.8·32-s + 183.·34-s − 145.·35-s + 154.·37-s + 45.2·38-s + 453.·40-s − 251.·41-s − 502.·43-s − 11.0·44-s + ⋯ |
| L(s) = 1 | − 0.969·2-s − 0.0604·4-s + 1.74·5-s − 0.404·7-s + 1.02·8-s − 1.68·10-s + 0.627·11-s + 0.391·14-s − 0.935·16-s − 0.956·17-s − 0.199·19-s − 0.105·20-s − 0.608·22-s + 1.59·23-s + 2.03·25-s + 0.0244·28-s − 1.86·29-s − 0.679·31-s − 0.120·32-s + 0.927·34-s − 0.704·35-s + 0.687·37-s + 0.193·38-s + 1.79·40-s − 0.958·41-s − 1.78·43-s − 0.0379·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.74T + 8T^{2} \) |
| 5 | \( 1 - 19.4T + 125T^{2} \) |
| 7 | \( 1 + 7.48T + 343T^{2} \) |
| 11 | \( 1 - 22.8T + 1.33e3T^{2} \) |
| 17 | \( 1 + 67.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 16.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 175.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 291.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 117.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 154.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 251.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 502.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 281.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 366.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 79.6T + 2.05e5T^{2} \) |
| 61 | \( 1 + 194.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 400.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 528.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 734.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 113.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 933.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.19e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 557.T + 9.12e5T^{2} \) |
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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
−9.195253064873447991330262613729, −8.168851354407513143474785003809, −6.94452044853434994534463211027, −6.51428847088952488083938286690, −5.43656342125336552568203560689, −4.69406347426726014261216827288, −3.33531280816128776210715032117, −2.01210570818027583291539427535, −1.39601899629840364230876764432, 0,
1.39601899629840364230876764432, 2.01210570818027583291539427535, 3.33531280816128776210715032117, 4.69406347426726014261216827288, 5.43656342125336552568203560689, 6.51428847088952488083938286690, 6.94452044853434994534463211027, 8.168851354407513143474785003809, 9.195253064873447991330262613729
