| L(s) = 1 | + 0.774·2-s − 15.2·3-s − 31.3·4-s − 11.7·6-s + 96.7·7-s − 49.1·8-s − 11.0·9-s − 121·11-s + 478.·12-s + 3.29·13-s + 74.9·14-s + 966.·16-s + 928.·17-s − 8.56·18-s + 1.42e3·19-s − 1.47e3·21-s − 93.7·22-s − 1.39e3·23-s + 747.·24-s + 2.55·26-s + 3.86e3·27-s − 3.03e3·28-s + 3.14e3·29-s + 4.29e3·31-s + 2.32e3·32-s + 1.84e3·33-s + 719.·34-s + ⋯ |
| L(s) = 1 | + 0.136·2-s − 0.976·3-s − 0.981·4-s − 0.133·6-s + 0.746·7-s − 0.271·8-s − 0.0454·9-s − 0.301·11-s + 0.958·12-s + 0.00540·13-s + 0.102·14-s + 0.944·16-s + 0.779·17-s − 0.00622·18-s + 0.906·19-s − 0.729·21-s − 0.0412·22-s − 0.549·23-s + 0.265·24-s + 0.000740·26-s + 1.02·27-s − 0.732·28-s + 0.694·29-s + 0.803·31-s + 0.400·32-s + 0.294·33-s + 0.106·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 - 0.774T + 32T^{2} \) |
| 3 | \( 1 + 15.2T + 243T^{2} \) |
| 7 | \( 1 - 96.7T + 1.68e4T^{2} \) |
| 13 | \( 1 - 3.29T + 3.71e5T^{2} \) |
| 17 | \( 1 - 928.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.42e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.39e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.14e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.29e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.38e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.30e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.15e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.07e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 6.15e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.65e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.17e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.22e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.00e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.04e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.75e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.97e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.23e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.51e4T + 8.58e9T^{2} \) |
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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
−10.56248204814157703440684565442, −9.788912656558466679542164761775, −8.550965837404751772833556721625, −7.77635174766147787358273604559, −6.27192751404447823856921270688, −5.27235516358426025532328357298, −4.71646721112779485244772302734, −3.23869133447979329762795176605, −1.22025234525268001398738966506, 0,
1.22025234525268001398738966506, 3.23869133447979329762795176605, 4.71646721112779485244772302734, 5.27235516358426025532328357298, 6.27192751404447823856921270688, 7.77635174766147787358273604559, 8.550965837404751772833556721625, 9.788912656558466679542164761775, 10.56248204814157703440684565442
