| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s − 5·5-s + 6·6-s − 28·7-s + 8·8-s + 9·9-s − 10·10-s − 36·11-s + 12·12-s + 13·13-s − 56·14-s − 15·15-s + 16·16-s + 42·17-s + 18·18-s − 112·19-s − 20·20-s − 84·21-s − 72·22-s − 168·23-s + 24·24-s + 25·25-s + 26·26-s + 27·27-s − 112·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.986·11-s + 0.288·12-s + 0.277·13-s − 1.06·14-s − 0.258·15-s + 1/4·16-s + 0.599·17-s + 0.235·18-s − 1.35·19-s − 0.223·20-s − 0.872·21-s − 0.697·22-s − 1.52·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 5 | \( 1 + p T \) |
| 13 | \( 1 - p T \) |
| good | 7 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 11 | \( 1 + 36 T + p^{3} T^{2} \) |
| 17 | \( 1 - 42 T + p^{3} T^{2} \) |
| 19 | \( 1 + 112 T + p^{3} T^{2} \) |
| 23 | \( 1 + 168 T + p^{3} T^{2} \) |
| 29 | \( 1 + 210 T + p^{3} T^{2} \) |
| 31 | \( 1 + 76 T + p^{3} T^{2} \) |
| 37 | \( 1 - 278 T + p^{3} T^{2} \) |
| 41 | \( 1 - 150 T + p^{3} T^{2} \) |
| 43 | \( 1 + 460 T + p^{3} T^{2} \) |
| 47 | \( 1 + 264 T + p^{3} T^{2} \) |
| 53 | \( 1 - 582 T + p^{3} T^{2} \) |
| 59 | \( 1 + 204 T + p^{3} T^{2} \) |
| 61 | \( 1 - 614 T + p^{3} T^{2} \) |
| 67 | \( 1 + 304 T + p^{3} T^{2} \) |
| 71 | \( 1 - 1080 T + p^{3} T^{2} \) |
| 73 | \( 1 + 934 T + p^{3} T^{2} \) |
| 79 | \( 1 - 128 T + p^{3} T^{2} \) |
| 83 | \( 1 - 348 T + p^{3} T^{2} \) |
| 89 | \( 1 + 834 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1582 T + p^{3} T^{2} \) |
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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
−10.34970176876581336820634933666, −9.700981604879443914816940732102, −8.441710174847126053909479781048, −7.57486060148700750452463852260, −6.52811221714818951388471894521, −5.63359416374899499585519206423, −4.13113128435342818801738871478, −3.37063427799304724298021889981, −2.26944481994306682534377430817, 0,
2.26944481994306682534377430817, 3.37063427799304724298021889981, 4.13113128435342818801738871478, 5.63359416374899499585519206423, 6.52811221714818951388471894521, 7.57486060148700750452463852260, 8.441710174847126053909479781048, 9.700981604879443914816940732102, 10.34970176876581336820634933666
