| L(s) = 1 | + 2-s − 1.56i·3-s + 4-s − i·5-s − 1.56i·6-s − i·7-s + 8-s + 0.561·9-s − i·10-s + 2i·11-s − 1.56i·12-s + 2.43·13-s − i·14-s − 1.56·15-s + 16-s − 4.12i·17-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.901i·3-s + 0.5·4-s − 0.447i·5-s − 0.637i·6-s − 0.377i·7-s + 0.353·8-s + 0.187·9-s − 0.316i·10-s + 0.603i·11-s − 0.450i·12-s + 0.676·13-s − 0.267i·14-s − 0.403·15-s + 0.250·16-s − 0.999i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.683098348\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.683098348\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + iT \) |
| 17 | \( 1 + 4.12iT \) |
| good | 3 | \( 1 + 1.56iT - 3T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 - 2.43T + 13T^{2} \) |
| 19 | \( 1 - 1.56T + 19T^{2} \) |
| 23 | \( 1 + 7.12iT - 23T^{2} \) |
| 29 | \( 1 - 6.43iT - 29T^{2} \) |
| 31 | \( 1 - 1.56iT - 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 2.87iT - 41T^{2} \) |
| 43 | \( 1 + 6.24T + 43T^{2} \) |
| 47 | \( 1 + 0.438T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 + 2.43T + 59T^{2} \) |
| 61 | \( 1 + 6.68iT - 61T^{2} \) |
| 67 | \( 1 - 7.12T + 67T^{2} \) |
| 71 | \( 1 - 0.438iT - 71T^{2} \) |
| 73 | \( 1 - 9.80iT - 73T^{2} \) |
| 79 | \( 1 - 7.36iT - 79T^{2} \) |
| 83 | \( 1 - 4.24T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 - 2.68iT - 97T^{2} \) |
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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
−9.600077614943311537745402126487, −8.545447126589585111640655644513, −7.71073991764904887577067147325, −6.90328403745287813033123980761, −6.42174824727889133241535113633, −5.14877067464327133446570787917, −4.49589534422908332852091281908, −3.33607653824191587358184691724, −2.08025894777975806749967251769, −0.993738097288097621508231177617,
1.72498835150788332891551979540, 3.22208925960849048035234144940, 3.75545479470995677164604421039, 4.73327024028700575291526488980, 5.74408023074737815990872906790, 6.29162296578601604915594667705, 7.46308055703956870485395691157, 8.304792724992196828723120445085, 9.357770080314364476386100415360, 10.01547203352004662924182551308
