| L(s) = 1 | − 5-s + 0.488i·7-s − 0.477·11-s − 4.91·13-s + 0.0686·17-s + 2.26i·19-s + (−4.64 + 1.19i)23-s + 25-s − 2.85i·29-s − 3.06·31-s − 0.488i·35-s − 5.16i·37-s + 6.04i·41-s − 2.17i·43-s − 0.440i·47-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.184i·7-s − 0.144·11-s − 1.36·13-s + 0.0166·17-s + 0.520i·19-s + (−0.968 + 0.248i)23-s + 0.200·25-s − 0.530i·29-s − 0.551·31-s − 0.0825i·35-s − 0.849i·37-s + 0.944i·41-s − 0.331i·43-s − 0.0642i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.111174255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.111174255\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + (4.64 - 1.19i)T \) |
| good | 7 | \( 1 - 0.488iT - 7T^{2} \) |
| 11 | \( 1 + 0.477T + 11T^{2} \) |
| 13 | \( 1 + 4.91T + 13T^{2} \) |
| 17 | \( 1 - 0.0686T + 17T^{2} \) |
| 19 | \( 1 - 2.26iT - 19T^{2} \) |
| 29 | \( 1 + 2.85iT - 29T^{2} \) |
| 31 | \( 1 + 3.06T + 31T^{2} \) |
| 37 | \( 1 + 5.16iT - 37T^{2} \) |
| 41 | \( 1 - 6.04iT - 41T^{2} \) |
| 43 | \( 1 + 2.17iT - 43T^{2} \) |
| 47 | \( 1 + 0.440iT - 47T^{2} \) |
| 53 | \( 1 + 5.95T + 53T^{2} \) |
| 59 | \( 1 + 2.25iT - 59T^{2} \) |
| 61 | \( 1 - 3.28iT - 61T^{2} \) |
| 67 | \( 1 - 6.36iT - 67T^{2} \) |
| 71 | \( 1 + 8.73iT - 71T^{2} \) |
| 73 | \( 1 - 13.5T + 73T^{2} \) |
| 79 | \( 1 + 9.13iT - 79T^{2} \) |
| 83 | \( 1 - 3.27T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 17.5iT - 97T^{2} \) |
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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
−7.76765905967278772338806178812, −7.23179285748256675716540814527, −6.39218181565758339005761854990, −5.65255389581893670016683547729, −4.97336893885644060373894331512, −4.21801118232791051453604720193, −3.50573289577107633972391006673, −2.55301000722412959860850860962, −1.84098425833737984608615112065, −0.42664289377587884559209230090,
0.58293642461789969053709212030, 1.91424636576097719461833841838, 2.70056500139585107487385955565, 3.56197154046437058079367778521, 4.37334997246589312610541138604, 4.98998697754038830301065634687, 5.68458788546723432827724469926, 6.67050745671930979282086819159, 7.14691034470990092680803166353, 7.84213589658825809429297445297
