| L(s) = 1 | − 3-s + 4·5-s + 4·7-s + 9-s − 4·15-s + 17-s − 4·21-s + 11·25-s − 27-s + 8·29-s + 8·31-s + 16·35-s + 5·37-s + 9·41-s + 2·43-s + 4·45-s + 6·47-s + 9·49-s − 51-s + 5·53-s − 5·59-s − 13·61-s + 4·63-s − 2·67-s − 9·71-s − 4·73-s − 11·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s + 1.51·7-s + 1/3·9-s − 1.03·15-s + 0.242·17-s − 0.872·21-s + 11/5·25-s − 0.192·27-s + 1.48·29-s + 1.43·31-s + 2.70·35-s + 0.821·37-s + 1.40·41-s + 0.304·43-s + 0.596·45-s + 0.875·47-s + 9/7·49-s − 0.140·51-s + 0.686·53-s − 0.650·59-s − 1.66·61-s + 0.503·63-s − 0.244·67-s − 1.06·71-s − 0.468·73-s − 1.27·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.284818541\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.284818541\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 13 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.67358344220245, −13.58496320686497, −12.69898045974273, −12.33981074416373, −11.80656777016641, −11.24051755590671, −10.76801165051872, −10.34204083194117, −9.938602287231324, −9.404208593397119, −8.762604880576758, −8.416684107228085, −7.670430257059287, −7.237299996308843, −6.472730350764181, −5.976553034598848, −5.731303285548498, −5.069022179181825, −4.487081798067207, −4.329836686121271, −2.900314662885910, −2.610634940492356, −1.826993858825699, −1.271543470187169, −0.8342713934118435,
0.8342713934118435, 1.271543470187169, 1.826993858825699, 2.610634940492356, 2.900314662885910, 4.329836686121271, 4.487081798067207, 5.069022179181825, 5.731303285548498, 5.976553034598848, 6.472730350764181, 7.237299996308843, 7.670430257059287, 8.416684107228085, 8.762604880576758, 9.404208593397119, 9.938602287231324, 10.34204083194117, 10.76801165051872, 11.24051755590671, 11.80656777016641, 12.33981074416373, 12.69898045974273, 13.58496320686497, 13.67358344220245
