| L(s) = 1 | + i·2-s + i·3-s − 4-s − 6-s + 2i·7-s − i·8-s − 9-s − i·12-s − 4i·13-s − 2·14-s + 16-s − i·17-s − i·18-s − 4·19-s − 2·21-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s + 0.755i·7-s − 0.353i·8-s − 0.333·9-s − 0.288i·12-s − 1.10i·13-s − 0.534·14-s + 0.250·16-s − 0.242i·17-s − 0.235i·18-s − 0.917·19-s − 0.436·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8610221119\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8610221119\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 17 | \( 1 + iT \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 + 10iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 - 2T + 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + 8iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−8.726612759448717761676608627680, −8.245713425672009827702196475572, −7.32700993895320354554923406000, −6.43136247515549689745410535311, −5.65613112164450728550952185048, −5.10887514205157063566702842108, −4.17811415120883478134440091476, −3.23468809516091379113423010245, −2.19495990950516027468695589797, −0.29829319085068413111911254106,
1.22127075589787683742109683507, 2.03246840653963387152554899326, 3.17237105256240963754999452597, 4.10377080365888073746864053291, 4.76779435809111368358280410886, 5.97354125526828108453193730752, 6.66347583389724343324811339243, 7.51253376315177692496404071230, 8.171645687220001427327486960866, 9.110788208929085661582988145167
