| L(s) = 1 | + 1.44·2-s + 2.24·3-s + 0.0881·4-s + 3.24·6-s − 1.35·7-s − 2.76·8-s + 2.04·9-s + 4.85·11-s + 0.198·12-s + 0.198·13-s − 1.96·14-s − 4.16·16-s + 1.13·17-s + 2.96·18-s − 3.04·21-s + 7.00·22-s − 2.55·23-s − 6.20·24-s + 0.286·26-s − 2.13·27-s − 0.119·28-s + 10.2·29-s − 2.51·31-s − 0.498·32-s + 10.8·33-s + 1.64·34-s + 0.180·36-s + ⋯ |
| L(s) = 1 | + 1.02·2-s + 1.29·3-s + 0.0440·4-s + 1.32·6-s − 0.512·7-s − 0.976·8-s + 0.682·9-s + 1.46·11-s + 0.0571·12-s + 0.0549·13-s − 0.524·14-s − 1.04·16-s + 0.275·17-s + 0.697·18-s − 0.665·21-s + 1.49·22-s − 0.532·23-s − 1.26·24-s + 0.0561·26-s − 0.411·27-s − 0.0226·28-s + 1.90·29-s − 0.451·31-s − 0.0880·32-s + 1.89·33-s + 0.281·34-s + 0.0301·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.138303622\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.138303622\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 1.44T + 2T^{2} \) |
| 3 | \( 1 - 2.24T + 3T^{2} \) |
| 7 | \( 1 + 1.35T + 7T^{2} \) |
| 11 | \( 1 - 4.85T + 11T^{2} \) |
| 13 | \( 1 - 0.198T + 13T^{2} \) |
| 17 | \( 1 - 1.13T + 17T^{2} \) |
| 23 | \( 1 + 2.55T + 23T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 + 2.51T + 31T^{2} \) |
| 37 | \( 1 + 0.137T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 - 7.59T + 43T^{2} \) |
| 47 | \( 1 + 2.69T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 + 5.82T + 59T^{2} \) |
| 61 | \( 1 + 7.58T + 61T^{2} \) |
| 67 | \( 1 - 8.01T + 67T^{2} \) |
| 71 | \( 1 - 8.82T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 + 3.77T + 83T^{2} \) |
| 89 | \( 1 + 9.36T + 89T^{2} \) |
| 97 | \( 1 - 0.198T + 97T^{2} \) |
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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
−7.76631157270599808473969548828, −6.93838738851322429718444509935, −6.24721475308029850437793454344, −5.75260015698829752360039326766, −4.64529806407585822543464903461, −4.07717210780626832911921360313, −3.53162473887435302649601916612, −2.90503724658975982384453740140, −2.16200728461287425340067204440, −0.889962719363712725182966712443,
0.889962719363712725182966712443, 2.16200728461287425340067204440, 2.90503724658975982384453740140, 3.53162473887435302649601916612, 4.07717210780626832911921360313, 4.64529806407585822543464903461, 5.75260015698829752360039326766, 6.24721475308029850437793454344, 6.93838738851322429718444509935, 7.76631157270599808473969548828
