| L(s) = 1 | − i·3-s − i·5-s + 4.59·7-s − 9-s − 4.88·11-s + 1.41·13-s − 15-s + 4.35i·17-s + 1.34·19-s − 4.59i·21-s + (−0.751 + 4.73i)23-s − 25-s + i·27-s − 8.47·29-s + 4.83i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.447i·5-s + 1.73·7-s − 0.333·9-s − 1.47·11-s + 0.392·13-s − 0.258·15-s + 1.05i·17-s + 0.307·19-s − 1.00i·21-s + (−0.156 + 0.987i)23-s − 0.200·25-s + 0.192i·27-s − 1.57·29-s + 0.867i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.192995523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.192995523\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + (0.751 - 4.73i)T \) |
| good | 7 | \( 1 - 4.59T + 7T^{2} \) |
| 11 | \( 1 + 4.88T + 11T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 - 4.35iT - 17T^{2} \) |
| 19 | \( 1 - 1.34T + 19T^{2} \) |
| 29 | \( 1 + 8.47T + 29T^{2} \) |
| 31 | \( 1 - 4.83iT - 31T^{2} \) |
| 37 | \( 1 - 2.55iT - 37T^{2} \) |
| 41 | \( 1 + 1.94T + 41T^{2} \) |
| 43 | \( 1 + 9.76T + 43T^{2} \) |
| 47 | \( 1 - 5.05iT - 47T^{2} \) |
| 53 | \( 1 - 11.0iT - 53T^{2} \) |
| 59 | \( 1 - 1.36iT - 59T^{2} \) |
| 61 | \( 1 - 9.11iT - 61T^{2} \) |
| 67 | \( 1 + 8.47T + 67T^{2} \) |
| 71 | \( 1 + 0.721iT - 71T^{2} \) |
| 73 | \( 1 - 2.29T + 73T^{2} \) |
| 79 | \( 1 - 15.2T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 + 8.99iT - 89T^{2} \) |
| 97 | \( 1 + 1.26iT - 97T^{2} \) |
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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
−8.217139826092499706518890286138, −7.71720603284511106465997142694, −7.18541354294162125202428782096, −5.93456851006759651386331409686, −5.42992166026671871440005287049, −4.85525098460740565833627051869, −3.95800011959185943959150368546, −2.88442136950807915147431026691, −1.75423763922538200921090383859, −1.37922795294303896356385013591,
0.28886490044777906332376277259, 1.85558284288887408858241288639, 2.54109392643136875126846710368, 3.54026210094628619713416743932, 4.42881448437065250021678148540, 5.30887651310007235681627552650, 5.33415842119718040992582283011, 6.62252206578077204852996747739, 7.50568803495836757960533253434, 8.008117017782723271751505239182
