| L(s) = 1 | − 2·2-s − 3·3-s − 4·4-s + 5·5-s + 6·6-s + 29·7-s + 24·8-s + 9·9-s − 10·10-s − 15·11-s + 12·12-s + 3·13-s − 58·14-s − 15·15-s − 16·16-s + 121·17-s − 18·18-s − 40·19-s − 20·20-s − 87·21-s + 30·22-s − 116·23-s − 72·24-s + 25·25-s − 6·26-s − 27·27-s − 116·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.56·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s − 0.411·11-s + 0.288·12-s + 0.0640·13-s − 1.10·14-s − 0.258·15-s − 1/4·16-s + 1.72·17-s − 0.235·18-s − 0.482·19-s − 0.223·20-s − 0.904·21-s + 0.290·22-s − 1.05·23-s − 0.612·24-s + 1/5·25-s − 0.0452·26-s − 0.192·27-s − 0.782·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.208537042\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.208537042\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| 29 | \( 1 - p T \) |
| good | 2 | \( 1 + p T + p^{3} T^{2} \) |
| 7 | \( 1 - 29 T + p^{3} T^{2} \) |
| 11 | \( 1 + 15 T + p^{3} T^{2} \) |
| 13 | \( 1 - 3 T + p^{3} T^{2} \) |
| 17 | \( 1 - 121 T + p^{3} T^{2} \) |
| 19 | \( 1 + 40 T + p^{3} T^{2} \) |
| 23 | \( 1 + 116 T + p^{3} T^{2} \) |
| 31 | \( 1 + 116 T + p^{3} T^{2} \) |
| 37 | \( 1 - 36 T + p^{3} T^{2} \) |
| 41 | \( 1 + 170 T + p^{3} T^{2} \) |
| 43 | \( 1 - 230 T + p^{3} T^{2} \) |
| 47 | \( 1 - 231 T + p^{3} T^{2} \) |
| 53 | \( 1 - 456 T + p^{3} T^{2} \) |
| 59 | \( 1 - 576 T + p^{3} T^{2} \) |
| 61 | \( 1 - 342 T + p^{3} T^{2} \) |
| 67 | \( 1 + 269 T + p^{3} T^{2} \) |
| 71 | \( 1 - 302 T + p^{3} T^{2} \) |
| 73 | \( 1 + 372 T + p^{3} T^{2} \) |
| 79 | \( 1 + 348 T + p^{3} T^{2} \) |
| 83 | \( 1 + 512 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1525 T + p^{3} T^{2} \) |
| 97 | \( 1 + 560 T + p^{3} T^{2} \) |
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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
−10.46015328944804877531754005646, −10.04326320608550751741039868651, −8.840299652756439931437108701791, −8.033446387775051280773096673024, −7.37465610209393064823993503875, −5.73572656962376573407477636334, −5.09728818457982808941375184185, −4.03776032702893288415155688609, −1.93210951186117854524521044168, −0.861620307838800919623260016528,
0.861620307838800919623260016528, 1.93210951186117854524521044168, 4.03776032702893288415155688609, 5.09728818457982808941375184185, 5.73572656962376573407477636334, 7.37465610209393064823993503875, 8.033446387775051280773096673024, 8.840299652756439931437108701791, 10.04326320608550751741039868651, 10.46015328944804877531754005646
