| L(s) = 1 | + 1.32·2-s − 0.239·4-s − 2.32·5-s − 7-s − 2.97·8-s − 3.08·10-s − 4.53·13-s − 1.32·14-s − 3.46·16-s − 3.29·17-s − 1.08·19-s + 0.556·20-s − 6.29·23-s + 0.414·25-s − 6.02·26-s + 0.239·28-s + 3.16·29-s − 2.97·31-s + 1.34·32-s − 4.37·34-s + 2.32·35-s + 1.70·37-s − 1.44·38-s + 6.91·40-s + 4.90·41-s − 8.38·43-s − 8.35·46-s + ⋯ |
| L(s) = 1 | + 0.938·2-s − 0.119·4-s − 1.04·5-s − 0.377·7-s − 1.05·8-s − 0.976·10-s − 1.25·13-s − 0.354·14-s − 0.866·16-s − 0.799·17-s − 0.249·19-s + 0.124·20-s − 1.31·23-s + 0.0829·25-s − 1.18·26-s + 0.0452·28-s + 0.587·29-s − 0.533·31-s + 0.237·32-s − 0.750·34-s + 0.393·35-s + 0.280·37-s − 0.234·38-s + 1.09·40-s + 0.766·41-s − 1.27·43-s − 1.23·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6230800437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6230800437\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.32T + 2T^{2} \) |
| 5 | \( 1 + 2.32T + 5T^{2} \) |
| 13 | \( 1 + 4.53T + 13T^{2} \) |
| 17 | \( 1 + 3.29T + 17T^{2} \) |
| 19 | \( 1 + 1.08T + 19T^{2} \) |
| 23 | \( 1 + 6.29T + 23T^{2} \) |
| 29 | \( 1 - 3.16T + 29T^{2} \) |
| 31 | \( 1 + 2.97T + 31T^{2} \) |
| 37 | \( 1 - 1.70T + 37T^{2} \) |
| 41 | \( 1 - 4.90T + 41T^{2} \) |
| 43 | \( 1 + 8.38T + 43T^{2} \) |
| 47 | \( 1 - 4.69T + 47T^{2} \) |
| 53 | \( 1 - 4.98T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 + 1.74T + 61T^{2} \) |
| 67 | \( 1 + 9.44T + 67T^{2} \) |
| 71 | \( 1 + 16.1T + 71T^{2} \) |
| 73 | \( 1 - 14.4T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 + 0.205T + 83T^{2} \) |
| 89 | \( 1 + 3.93T + 89T^{2} \) |
| 97 | \( 1 - 13.0T + 97T^{2} \) |
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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
−7.71347689065683401629090824469, −7.20417234833928040548436391035, −6.30501518938440447080368126842, −5.76397134124832525549268583389, −4.72802026302015625126871834626, −4.42067479445041096948942444673, −3.70203626875090808629778713414, −2.95678964609476940120315718375, −2.10943688428549132221393850007, −0.31936397473343321203614815351,
0.31936397473343321203614815351, 2.10943688428549132221393850007, 2.95678964609476940120315718375, 3.70203626875090808629778713414, 4.42067479445041096948942444673, 4.72802026302015625126871834626, 5.76397134124832525549268583389, 6.30501518938440447080368126842, 7.20417234833928040548436391035, 7.71347689065683401629090824469
