| L(s) = 1 | − 2-s + 1.23·3-s + 4-s − 2.85·5-s − 1.23·6-s − 2.85·7-s − 8-s − 1.47·9-s + 2.85·10-s + 1.23·12-s + 4·13-s + 2.85·14-s − 3.52·15-s + 16-s + 0.618·17-s + 1.47·18-s − 19-s − 2.85·20-s − 3.52·21-s − 5.85·23-s − 1.23·24-s + 3.14·25-s − 4·26-s − 5.52·27-s − 2.85·28-s − 7.70·29-s + 3.52·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.713·3-s + 0.5·4-s − 1.27·5-s − 0.504·6-s − 1.07·7-s − 0.353·8-s − 0.490·9-s + 0.902·10-s + 0.356·12-s + 1.10·13-s + 0.762·14-s − 0.910·15-s + 0.250·16-s + 0.149·17-s + 0.346·18-s − 0.229·19-s − 0.638·20-s − 0.769·21-s − 1.22·23-s − 0.252·24-s + 0.629·25-s − 0.784·26-s − 1.06·27-s − 0.539·28-s − 1.43·29-s + 0.644·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6391704865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6391704865\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 1.23T + 3T^{2} \) |
| 5 | \( 1 + 2.85T + 5T^{2} \) |
| 7 | \( 1 + 2.85T + 7T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 - 0.618T + 17T^{2} \) |
| 23 | \( 1 + 5.85T + 23T^{2} \) |
| 29 | \( 1 + 7.70T + 29T^{2} \) |
| 31 | \( 1 + 7.23T + 31T^{2} \) |
| 37 | \( 1 - 1.23T + 37T^{2} \) |
| 41 | \( 1 - 8.47T + 41T^{2} \) |
| 43 | \( 1 - 5.38T + 43T^{2} \) |
| 47 | \( 1 - 5.85T + 47T^{2} \) |
| 53 | \( 1 - 6.76T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 - 7.85T + 61T^{2} \) |
| 67 | \( 1 + 4.76T + 67T^{2} \) |
| 71 | \( 1 + 7.23T + 71T^{2} \) |
| 73 | \( 1 + 6.94T + 73T^{2} \) |
| 79 | \( 1 - 4.76T + 79T^{2} \) |
| 83 | \( 1 + 8.85T + 83T^{2} \) |
| 89 | \( 1 - 8.18T + 89T^{2} \) |
| 97 | \( 1 + 0.944T + 97T^{2} \) |
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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
−8.374508146580116201267514327885, −7.60308238571777102617995221749, −7.33149053428695187455108117250, −6.10763884340303475951550661002, −5.77489541493939785186379776921, −4.09041213427590088403324883838, −3.69191689460986111245311731807, −2.98854530017053091426717153503, −1.96173446325727912843908943916, −0.45341238646763227297945922383,
0.45341238646763227297945922383, 1.96173446325727912843908943916, 2.98854530017053091426717153503, 3.69191689460986111245311731807, 4.09041213427590088403324883838, 5.77489541493939785186379776921, 6.10763884340303475951550661002, 7.33149053428695187455108117250, 7.60308238571777102617995221749, 8.374508146580116201267514327885
