| L(s) = 1 | − 2.43·2-s + 3.91·4-s − 4.51·7-s − 4.66·8-s + 5.03·11-s + 13-s + 10.9·14-s + 3.51·16-s + 2.06·17-s − 2.81·19-s − 12.2·22-s − 4.04·23-s − 2.43·26-s − 17.6·28-s − 7.83·31-s + 0.788·32-s − 5.02·34-s + 6.34·37-s + 6.84·38-s + 1.47·41-s − 6.81·43-s + 19.7·44-s + 9.83·46-s + 9.07·47-s + 13.3·49-s + 3.91·52-s + 9.81·53-s + ⋯ |
| L(s) = 1 | − 1.72·2-s + 1.95·4-s − 1.70·7-s − 1.64·8-s + 1.51·11-s + 0.277·13-s + 2.93·14-s + 0.877·16-s + 0.500·17-s − 0.645·19-s − 2.60·22-s − 0.843·23-s − 0.477·26-s − 3.33·28-s − 1.40·31-s + 0.139·32-s − 0.861·34-s + 1.04·37-s + 1.11·38-s + 0.230·41-s − 1.03·43-s + 2.97·44-s + 1.45·46-s + 1.32·47-s + 1.90·49-s + 0.543·52-s + 1.34·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 2.43T + 2T^{2} \) |
| 7 | \( 1 + 4.51T + 7T^{2} \) |
| 11 | \( 1 - 5.03T + 11T^{2} \) |
| 17 | \( 1 - 2.06T + 17T^{2} \) |
| 19 | \( 1 + 2.81T + 19T^{2} \) |
| 23 | \( 1 + 4.04T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 7.83T + 31T^{2} \) |
| 37 | \( 1 - 6.34T + 37T^{2} \) |
| 41 | \( 1 - 1.47T + 41T^{2} \) |
| 43 | \( 1 + 6.81T + 43T^{2} \) |
| 47 | \( 1 - 9.07T + 47T^{2} \) |
| 53 | \( 1 - 9.81T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 - 8.34T + 61T^{2} \) |
| 67 | \( 1 + 15.8T + 67T^{2} \) |
| 71 | \( 1 - 0.569T + 71T^{2} \) |
| 73 | \( 1 + 0.373T + 73T^{2} \) |
| 79 | \( 1 - 6.13T + 79T^{2} \) |
| 83 | \( 1 + 0.654T + 83T^{2} \) |
| 89 | \( 1 + 5.93T + 89T^{2} \) |
| 97 | \( 1 + 2.67T + 97T^{2} \) |
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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
−8.765399692830817663482389570905, −7.66407121217256934680498578785, −7.02959048809068781218756516919, −6.32289570009938423808623580511, −5.90995091188175326492924182983, −4.12099102756352109305240865550, −3.38177310889204609098186919212, −2.27500786299593598773464259603, −1.16305420820047936214787808890, 0,
1.16305420820047936214787808890, 2.27500786299593598773464259603, 3.38177310889204609098186919212, 4.12099102756352109305240865550, 5.90995091188175326492924182983, 6.32289570009938423808623580511, 7.02959048809068781218756516919, 7.66407121217256934680498578785, 8.765399692830817663482389570905
