L(s) = 1 | − 0.249·2-s + 2.30·3-s − 1.93·4-s − 0.574·6-s − 3.96·7-s + 0.983·8-s + 2.29·9-s + 3.13·11-s − 4.45·12-s − 2.65·13-s + 0.989·14-s + 3.62·16-s − 2.25·17-s − 0.573·18-s − 9.11·21-s − 0.782·22-s + 7.58·23-s + 2.26·24-s + 0.664·26-s − 1.61·27-s + 7.67·28-s − 1.36·29-s − 0.894·31-s − 2.87·32-s + 7.21·33-s + 0.563·34-s − 4.45·36-s + ⋯ |
L(s) = 1 | − 0.176·2-s + 1.32·3-s − 0.968·4-s − 0.234·6-s − 1.49·7-s + 0.347·8-s + 0.765·9-s + 0.945·11-s − 1.28·12-s − 0.737·13-s + 0.264·14-s + 0.907·16-s − 0.547·17-s − 0.135·18-s − 1.98·21-s − 0.166·22-s + 1.58·23-s + 0.462·24-s + 0.130·26-s − 0.311·27-s + 1.45·28-s − 0.253·29-s − 0.160·31-s − 0.507·32-s + 1.25·33-s + 0.0966·34-s − 0.741·36-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + 0.249T + 2T^{2} \) |
| 3 | \( 1 - 2.30T + 3T^{2} \) |
| 7 | \( 1 + 3.96T + 7T^{2} \) |
| 11 | \( 1 - 3.13T + 11T^{2} \) |
| 13 | \( 1 + 2.65T + 13T^{2} \) |
| 17 | \( 1 + 2.25T + 17T^{2} \) |
| 23 | \( 1 - 7.58T + 23T^{2} \) |
| 29 | \( 1 + 1.36T + 29T^{2} \) |
| 31 | \( 1 + 0.894T + 31T^{2} \) |
| 37 | \( 1 - 6.62T + 37T^{2} \) |
| 41 | \( 1 + 6.23T + 41T^{2} \) |
| 43 | \( 1 - 1.77T + 43T^{2} \) |
| 47 | \( 1 + 0.176T + 47T^{2} \) |
| 53 | \( 1 - 6.77T + 53T^{2} \) |
| 59 | \( 1 - 7.81T + 59T^{2} \) |
| 61 | \( 1 + 1.03T + 61T^{2} \) |
| 67 | \( 1 - 15.2T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 4.18T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 + 1.48T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
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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
−7.29293253789367587457748880952, −7.03425176737864882486303679928, −6.10373975051800525629705487860, −5.24505271086401227581770139178, −4.30152679842994858881273719523, −3.76292723673140757324353587481, −3.08519012363534867040524372164, −2.45951779584071871528083863508, −1.19346345061662421092177743042, 0,
1.19346345061662421092177743042, 2.45951779584071871528083863508, 3.08519012363534867040524372164, 3.76292723673140757324353587481, 4.30152679842994858881273719523, 5.24505271086401227581770139178, 6.10373975051800525629705487860, 7.03425176737864882486303679928, 7.29293253789367587457748880952