| L(s) = 1 | − 0.421i·3-s − 5-s + (−1.02 + 2.43i)7-s + 2.82·9-s + 1.89·11-s − 4.65·13-s + 0.421i·15-s + 7.19i·17-s + 0.834i·19-s + (1.02 + 0.432i)21-s − 5.76i·23-s + 25-s − 2.45i·27-s − 4.84i·29-s − 4.04·31-s + ⋯ |
| L(s) = 1 | − 0.243i·3-s − 0.447·5-s + (−0.387 + 0.921i)7-s + 0.940·9-s + 0.572·11-s − 1.29·13-s + 0.108i·15-s + 1.74i·17-s + 0.191i·19-s + (0.224 + 0.0943i)21-s − 1.20i·23-s + 0.200·25-s − 0.472i·27-s − 0.899i·29-s − 0.726·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6593543404\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6593543404\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (1.02 - 2.43i)T \) |
| good | 3 | \( 1 + 0.421iT - 3T^{2} \) |
| 11 | \( 1 - 1.89T + 11T^{2} \) |
| 13 | \( 1 + 4.65T + 13T^{2} \) |
| 17 | \( 1 - 7.19iT - 17T^{2} \) |
| 19 | \( 1 - 0.834iT - 19T^{2} \) |
| 23 | \( 1 + 5.76iT - 23T^{2} \) |
| 29 | \( 1 + 4.84iT - 29T^{2} \) |
| 31 | \( 1 + 4.04T + 31T^{2} \) |
| 37 | \( 1 - 4.68iT - 37T^{2} \) |
| 41 | \( 1 - 5.81iT - 41T^{2} \) |
| 43 | \( 1 + 12.6T + 43T^{2} \) |
| 47 | \( 1 + 6.30T + 47T^{2} \) |
| 53 | \( 1 - 0.637iT - 53T^{2} \) |
| 59 | \( 1 + 5.32iT - 59T^{2} \) |
| 61 | \( 1 - 7.80T + 61T^{2} \) |
| 67 | \( 1 + 4.64T + 67T^{2} \) |
| 71 | \( 1 - 6.42iT - 71T^{2} \) |
| 73 | \( 1 + 0.227iT - 73T^{2} \) |
| 79 | \( 1 - 7.17iT - 79T^{2} \) |
| 83 | \( 1 - 4.76iT - 83T^{2} \) |
| 89 | \( 1 - 11.9iT - 89T^{2} \) |
| 97 | \( 1 - 10.3iT - 97T^{2} \) |
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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
−9.479084451336862025134597925533, −8.291787830856369181604766546781, −8.062325321714209811514215174728, −6.72154798075682732995156250941, −6.55744856154242941393727940451, −5.37611947221384208575450701084, −4.46654917506249095385386298740, −3.68179950564887074708942783652, −2.51164723305557337322928112259, −1.55645849589378746789047662353,
0.22508139355954959628852882219, 1.60091122818339108383983196643, 3.06473839890215775062753165176, 3.81817612322282869860888188455, 4.69307529689726419057500621579, 5.29315969593075017861462693487, 6.78921530875870044214765562737, 7.19705862824505315545947217184, 7.61677781332156900465974873140, 8.944748292505353199334348378607
