| L(s) = 1 | − 2.80·2-s + 5.84·4-s − 1.04·7-s − 10.7·8-s − 4.64·11-s − 2.95·13-s + 2.91·14-s + 18.4·16-s + 6.29·17-s − 1.11·19-s + 12.9·22-s + 3.87·23-s + 8.28·26-s − 6.08·28-s − 2.35·29-s + 31-s − 30.1·32-s − 17.6·34-s − 1.30·37-s + 3.12·38-s − 1.92·41-s + 4.29·43-s − 27.1·44-s − 10.8·46-s − 3.31·47-s − 5.91·49-s − 17.2·52-s + ⋯ |
| L(s) = 1 | − 1.98·2-s + 2.92·4-s − 0.393·7-s − 3.80·8-s − 1.39·11-s − 0.820·13-s + 0.779·14-s + 4.61·16-s + 1.52·17-s − 0.255·19-s + 2.77·22-s + 0.807·23-s + 1.62·26-s − 1.14·28-s − 0.437·29-s + 0.179·31-s − 5.32·32-s − 3.02·34-s − 0.213·37-s + 0.506·38-s − 0.300·41-s + 0.655·43-s − 4.08·44-s − 1.59·46-s − 0.483·47-s − 0.845·49-s − 2.39·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 2.80T + 2T^{2} \) |
| 7 | \( 1 + 1.04T + 7T^{2} \) |
| 11 | \( 1 + 4.64T + 11T^{2} \) |
| 13 | \( 1 + 2.95T + 13T^{2} \) |
| 17 | \( 1 - 6.29T + 17T^{2} \) |
| 19 | \( 1 + 1.11T + 19T^{2} \) |
| 23 | \( 1 - 3.87T + 23T^{2} \) |
| 29 | \( 1 + 2.35T + 29T^{2} \) |
| 37 | \( 1 + 1.30T + 37T^{2} \) |
| 41 | \( 1 + 1.92T + 41T^{2} \) |
| 43 | \( 1 - 4.29T + 43T^{2} \) |
| 47 | \( 1 + 3.31T + 47T^{2} \) |
| 53 | \( 1 - 5.10T + 53T^{2} \) |
| 59 | \( 1 - 5.07T + 59T^{2} \) |
| 61 | \( 1 - 5.31T + 61T^{2} \) |
| 67 | \( 1 - 8.64T + 67T^{2} \) |
| 71 | \( 1 - 3.88T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 + 7.12T + 79T^{2} \) |
| 83 | \( 1 + 7.46T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 - 0.915T + 97T^{2} \) |
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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
−7.60286199583003591172750913035, −7.35884684493297232430904375285, −6.52892816843157504425426167425, −5.69959854778107121747650483994, −5.10061848210629678275539320240, −3.45275633960063556442038622315, −2.79945199396864937014526331250, −2.10126970126356970023772174219, −0.967685613348648057397485017879, 0,
0.967685613348648057397485017879, 2.10126970126356970023772174219, 2.79945199396864937014526331250, 3.45275633960063556442038622315, 5.10061848210629678275539320240, 5.69959854778107121747650483994, 6.52892816843157504425426167425, 7.35884684493297232430904375285, 7.60286199583003591172750913035
