| L(s) = 1 | − i·3-s + 2.12·5-s + (−2.43 − 1.03i)7-s − 9-s + 1.02i·11-s + 1.76i·13-s − 2.12i·15-s − 0.210·17-s + 4.64·19-s + (−1.03 + 2.43i)21-s + (3.56 − 3.20i)23-s − 0.474·25-s + i·27-s + 10.2·29-s − 1.19i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.951·5-s + (−0.919 − 0.392i)7-s − 0.333·9-s + 0.308i·11-s + 0.488i·13-s − 0.549i·15-s − 0.0510·17-s + 1.06·19-s + (−0.226 + 0.531i)21-s + (0.742 − 0.669i)23-s − 0.0948·25-s + 0.192i·27-s + 1.89·29-s − 0.213i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.324 + 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.324 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.818792392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.818792392\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 + (2.43 + 1.03i)T \) |
| 23 | \( 1 + (-3.56 + 3.20i)T \) |
| good | 5 | \( 1 - 2.12T + 5T^{2} \) |
| 11 | \( 1 - 1.02iT - 11T^{2} \) |
| 13 | \( 1 - 1.76iT - 13T^{2} \) |
| 17 | \( 1 + 0.210T + 17T^{2} \) |
| 19 | \( 1 - 4.64T + 19T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 + 1.19iT - 31T^{2} \) |
| 37 | \( 1 + 10.7iT - 37T^{2} \) |
| 41 | \( 1 + 1.14iT - 41T^{2} \) |
| 43 | \( 1 + 9.64iT - 43T^{2} \) |
| 47 | \( 1 + 5.56iT - 47T^{2} \) |
| 53 | \( 1 - 1.32iT - 53T^{2} \) |
| 59 | \( 1 + 9.90iT - 59T^{2} \) |
| 61 | \( 1 + 2.05T + 61T^{2} \) |
| 67 | \( 1 + 2.73iT - 67T^{2} \) |
| 71 | \( 1 - 6.57T + 71T^{2} \) |
| 73 | \( 1 - 12.6iT - 73T^{2} \) |
| 79 | \( 1 - 1.45iT - 79T^{2} \) |
| 83 | \( 1 + 6.66T + 83T^{2} \) |
| 89 | \( 1 + 6.85T + 89T^{2} \) |
| 97 | \( 1 + 1.67T + 97T^{2} \) |
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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
−9.163568538354988869253564991504, −8.311006613188276133406711323148, −7.14726811939975966092980420356, −6.81303532457657167360820506830, −5.94873991149967876248330657000, −5.18813703392498201527178348055, −4.01481775983277948474278074426, −2.91122832283362891182182454472, −2.03374485886978102964643762538, −0.74090415727601504534216559662,
1.20482435359114051848830325581, 2.83785525778304384077429982931, 3.16649295091273909585065394746, 4.58261966663506877446905911562, 5.41466397359378776102883633631, 6.08182150423551029951598735088, 6.76016600132455650629792479558, 7.917739606771665325662700581532, 8.767174184366003241083675653045, 9.564882341761098624088567123166
