| L(s) = 1 | − 2.63i·5-s + 7-s + 4.29i·11-s − 0.0480i·13-s − 3.16·17-s + 0.726i·19-s + 5.28·23-s − 1.94·25-s − 8.00i·29-s − 4.90·31-s − 2.63i·35-s + 9.10i·37-s + 1.45·41-s − 8.85i·43-s + 9.23·47-s + ⋯ |
| L(s) = 1 | − 1.17i·5-s + 0.377·7-s + 1.29i·11-s − 0.0133i·13-s − 0.766·17-s + 0.166i·19-s + 1.10·23-s − 0.388·25-s − 1.48i·29-s − 0.881·31-s − 0.445i·35-s + 1.49i·37-s + 0.227·41-s − 1.35i·43-s + 1.34·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.587 + 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.587 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.933607365\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.933607365\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + 2.63iT - 5T^{2} \) |
| 11 | \( 1 - 4.29iT - 11T^{2} \) |
| 13 | \( 1 + 0.0480iT - 13T^{2} \) |
| 17 | \( 1 + 3.16T + 17T^{2} \) |
| 19 | \( 1 - 0.726iT - 19T^{2} \) |
| 23 | \( 1 - 5.28T + 23T^{2} \) |
| 29 | \( 1 + 8.00iT - 29T^{2} \) |
| 31 | \( 1 + 4.90T + 31T^{2} \) |
| 37 | \( 1 - 9.10iT - 37T^{2} \) |
| 41 | \( 1 - 1.45T + 41T^{2} \) |
| 43 | \( 1 + 8.85iT - 43T^{2} \) |
| 47 | \( 1 - 9.23T + 47T^{2} \) |
| 53 | \( 1 + 3.14iT - 53T^{2} \) |
| 59 | \( 1 + 5.90iT - 59T^{2} \) |
| 61 | \( 1 + 2.19iT - 61T^{2} \) |
| 67 | \( 1 - 8.55iT - 67T^{2} \) |
| 71 | \( 1 - 3.86T + 71T^{2} \) |
| 73 | \( 1 + 7.56T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 + 9.05iT - 83T^{2} \) |
| 89 | \( 1 - 5.15T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−7.991173693039918268774770783969, −7.32342298232592278767079984524, −6.64027833496208342031261057381, −5.67726852146481969318474425378, −4.89356013154462863129169650859, −4.55841922480646555105894454746, −3.71902420633902050057702321309, −2.40926873537918412645754527069, −1.70121632273132490629122292087, −0.62530514474568575639686401884,
0.876711664347888639587491810701, 2.15385471148641627465579900792, 2.99981787180300132351534055025, 3.54526902663940787990047824498, 4.53882948834316772041299644418, 5.44644325124617902524892128608, 6.07758522499235668243394672665, 6.88186577799014727564045084147, 7.31101600475490714252831665879, 8.130826762916214126314952714980
