| L(s) = 1 | + 2.77·2-s + 7.63·3-s − 0.318·4-s − 4.52·5-s + 21.1·6-s − 2.79·7-s − 23.0·8-s + 31.2·9-s − 12.5·10-s − 2.42·12-s − 13·13-s − 7.74·14-s − 34.5·15-s − 61.3·16-s + 56.1·17-s + 86.6·18-s − 40.8·19-s + 1.43·20-s − 21.3·21-s + 83.8·23-s − 175.·24-s − 104.·25-s − 36.0·26-s + 32.5·27-s + 0.888·28-s + 22.9·29-s − 95.6·30-s + ⋯ |
| L(s) = 1 | + 0.979·2-s + 1.46·3-s − 0.0397·4-s − 0.404·5-s + 1.43·6-s − 0.150·7-s − 1.01·8-s + 1.15·9-s − 0.396·10-s − 0.0584·12-s − 0.277·13-s − 0.147·14-s − 0.593·15-s − 0.958·16-s + 0.801·17-s + 1.13·18-s − 0.492·19-s + 0.0160·20-s − 0.221·21-s + 0.760·23-s − 1.49·24-s − 0.836·25-s − 0.271·26-s + 0.231·27-s + 0.00599·28-s + 0.146·29-s − 0.582·30-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 | \( 1 - 2.77T + 8T^{2} \) |
| 3 | \( 1 - 7.63T + 27T^{2} \) |
| 5 | \( 1 + 4.52T + 125T^{2} \) |
| 7 | \( 1 + 2.79T + 343T^{2} \) |
| 17 | \( 1 - 56.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 40.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 83.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 22.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 199.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 47.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 50.5T + 6.89e4T^{2} \) |
| 43 | \( 1 + 330.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 387.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 307.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 78.1T + 2.05e5T^{2} \) |
| 61 | \( 1 + 430.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 393.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 379.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 87.7T + 3.89e5T^{2} \) |
| 79 | \( 1 + 787.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 597.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 998.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.59e3T + 9.12e5T^{2} \) |
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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
−8.591823469260922655362865135711, −7.978047917303866535208061261105, −7.15587126614569186288070121416, −6.11684819815659462765041338121, −5.10735710190527023782455187962, −4.24837449187021322856686236478, −3.42482063576983757656193190524, −2.96875437608898878339454437477, −1.76898642062672795525945250793, 0,
1.76898642062672795525945250793, 2.96875437608898878339454437477, 3.42482063576983757656193190524, 4.24837449187021322856686236478, 5.10735710190527023782455187962, 6.11684819815659462765041338121, 7.15587126614569186288070121416, 7.978047917303866535208061261105, 8.591823469260922655362865135711
