| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 0.929·7-s + 8-s + 9-s − 1.29·11-s + 12-s + 0.581·13-s + 0.929·14-s + 16-s + 0.338·17-s + 18-s − 1.27·19-s + 0.929·21-s − 1.29·22-s + 0.403·23-s + 24-s + 0.581·26-s + 27-s + 0.929·28-s + 8.82·29-s + 5.29·31-s + 32-s − 1.29·33-s + 0.338·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s + 0.351·7-s + 0.353·8-s + 0.333·9-s − 0.390·11-s + 0.288·12-s + 0.161·13-s + 0.248·14-s + 0.250·16-s + 0.0820·17-s + 0.235·18-s − 0.293·19-s + 0.202·21-s − 0.276·22-s + 0.0841·23-s + 0.204·24-s + 0.114·26-s + 0.192·27-s + 0.175·28-s + 1.63·29-s + 0.950·31-s + 0.176·32-s − 0.225·33-s + 0.0580·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.080877673\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.080877673\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 0.929T + 7T^{2} \) |
| 11 | \( 1 + 1.29T + 11T^{2} \) |
| 13 | \( 1 - 0.581T + 13T^{2} \) |
| 17 | \( 1 - 0.338T + 17T^{2} \) |
| 19 | \( 1 + 1.27T + 19T^{2} \) |
| 23 | \( 1 - 0.403T + 23T^{2} \) |
| 29 | \( 1 - 8.82T + 29T^{2} \) |
| 31 | \( 1 - 5.29T + 31T^{2} \) |
| 37 | \( 1 - 3.51T + 37T^{2} \) |
| 41 | \( 1 - 3.05T + 41T^{2} \) |
| 43 | \( 1 - 4.16T + 43T^{2} \) |
| 47 | \( 1 + 7.68T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 - 7.83T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 - 7.97T + 67T^{2} \) |
| 71 | \( 1 - 8.54T + 71T^{2} \) |
| 73 | \( 1 - 3.68T + 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 - 2.45T + 83T^{2} \) |
| 89 | \( 1 + 8.97T + 89T^{2} \) |
| 97 | \( 1 + 18.9T + 97T^{2} \) |
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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
−8.171480466878935991491583689244, −8.015040775045239728259798156620, −6.89471796549311376899027549986, −6.35439666086411747990446097365, −5.36138301624236024092494538186, −4.64881594726560737127305854195, −3.94120912820747692065226998192, −2.94841463796701756914533878385, −2.30711202182688657394638661980, −1.08884274022129004986707438654,
1.08884274022129004986707438654, 2.30711202182688657394638661980, 2.94841463796701756914533878385, 3.94120912820747692065226998192, 4.64881594726560737127305854195, 5.36138301624236024092494538186, 6.35439666086411747990446097365, 6.89471796549311376899027549986, 8.015040775045239728259798156620, 8.171480466878935991491583689244
