| L(s) = 1 | + 2·2-s + 4·4-s + 9·5-s − 31·7-s + 8·8-s + 18·10-s − 57·11-s − 52·13-s − 62·14-s + 16·16-s − 69·17-s + 19·19-s + 36·20-s − 114·22-s + 72·23-s − 44·25-s − 104·26-s − 124·28-s + 150·29-s + 32·31-s + 32·32-s − 138·34-s − 279·35-s − 226·37-s + 38·38-s + 72·40-s + 258·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.804·5-s − 1.67·7-s + 0.353·8-s + 0.569·10-s − 1.56·11-s − 1.10·13-s − 1.18·14-s + 1/4·16-s − 0.984·17-s + 0.229·19-s + 0.402·20-s − 1.10·22-s + 0.652·23-s − 0.351·25-s − 0.784·26-s − 0.836·28-s + 0.960·29-s + 0.185·31-s + 0.176·32-s − 0.696·34-s − 1.34·35-s − 1.00·37-s + 0.162·38-s + 0.284·40-s + 0.982·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 19 | \( 1 - p T \) |
| good | 5 | \( 1 - 9 T + p^{3} T^{2} \) |
| 7 | \( 1 + 31 T + p^{3} T^{2} \) |
| 11 | \( 1 + 57 T + p^{3} T^{2} \) |
| 13 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 17 | \( 1 + 69 T + p^{3} T^{2} \) |
| 23 | \( 1 - 72 T + p^{3} T^{2} \) |
| 29 | \( 1 - 150 T + p^{3} T^{2} \) |
| 31 | \( 1 - 32 T + p^{3} T^{2} \) |
| 37 | \( 1 + 226 T + p^{3} T^{2} \) |
| 41 | \( 1 - 258 T + p^{3} T^{2} \) |
| 43 | \( 1 + 67 T + p^{3} T^{2} \) |
| 47 | \( 1 + 579 T + p^{3} T^{2} \) |
| 53 | \( 1 - 432 T + p^{3} T^{2} \) |
| 59 | \( 1 - 330 T + p^{3} T^{2} \) |
| 61 | \( 1 + 13 T + p^{3} T^{2} \) |
| 67 | \( 1 + 856 T + p^{3} T^{2} \) |
| 71 | \( 1 + 642 T + p^{3} T^{2} \) |
| 73 | \( 1 + 487 T + p^{3} T^{2} \) |
| 79 | \( 1 + 700 T + p^{3} T^{2} \) |
| 83 | \( 1 - 12 T + p^{3} T^{2} \) |
| 89 | \( 1 - 600 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1424 T + p^{3} T^{2} \) |
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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
−10.34341847202339848520724765005, −10.03494671085738858954730951294, −8.934449254796092156719722387071, −7.44872061679010872930925968877, −6.58791117251959824723248135185, −5.68035013070033519135946723234, −4.71867965872099168335026473277, −3.10280530137574496245859509684, −2.37274408143447097467806081446, 0,
2.37274408143447097467806081446, 3.10280530137574496245859509684, 4.71867965872099168335026473277, 5.68035013070033519135946723234, 6.58791117251959824723248135185, 7.44872061679010872930925968877, 8.934449254796092156719722387071, 10.03494671085738858954730951294, 10.34341847202339848520724765005
