| L(s) = 1 | + 1.14·2-s − 1.98·3-s − 6.69·4-s + 4.19·5-s − 2.25·6-s − 2.88·7-s − 16.7·8-s − 23.0·9-s + 4.77·10-s + 13.2·12-s + 13·13-s − 3.29·14-s − 8.29·15-s + 34.4·16-s − 32.5·17-s − 26.3·18-s + 69.3·19-s − 28.0·20-s + 5.71·21-s + 146.·23-s + 33.1·24-s − 107.·25-s + 14.8·26-s + 99.1·27-s + 19.3·28-s + 187.·29-s − 9.46·30-s + ⋯ |
| L(s) = 1 | + 0.403·2-s − 0.381·3-s − 0.837·4-s + 0.374·5-s − 0.153·6-s − 0.155·7-s − 0.740·8-s − 0.854·9-s + 0.151·10-s + 0.319·12-s + 0.277·13-s − 0.0628·14-s − 0.142·15-s + 0.538·16-s − 0.463·17-s − 0.344·18-s + 0.837·19-s − 0.313·20-s + 0.0593·21-s + 1.32·23-s + 0.282·24-s − 0.859·25-s + 0.111·26-s + 0.706·27-s + 0.130·28-s + 1.19·29-s − 0.0575·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 2 | \( 1 - 1.14T + 8T^{2} \) |
| 3 | \( 1 + 1.98T + 27T^{2} \) |
| 5 | \( 1 - 4.19T + 125T^{2} \) |
| 7 | \( 1 + 2.88T + 343T^{2} \) |
| 17 | \( 1 + 32.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 69.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 146.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 187.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 271.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 225.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 51.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 415.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 237.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 442.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 661.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 581.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 683.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.13e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 267.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.10e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 331.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.01e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 181.T + 9.12e5T^{2} \) |
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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
−8.662816650796589098583123100048, −8.101681490877711649961566378081, −6.74789127007093907686738044301, −6.11496879867697866731213844168, −5.21534655869035147456419397077, −4.73623867968681955955501581100, −3.49013181303573439053524469679, −2.75243190348867659410996769440, −1.12722615361679437978161071332, 0,
1.12722615361679437978161071332, 2.75243190348867659410996769440, 3.49013181303573439053524469679, 4.73623867968681955955501581100, 5.21534655869035147456419397077, 6.11496879867697866731213844168, 6.74789127007093907686738044301, 8.101681490877711649961566378081, 8.662816650796589098583123100048
