| L(s) = 1 | − 2-s + 3-s − 4-s + 5-s − 6-s + 7-s + 3·8-s + 9-s − 10-s − 4·11-s − 12-s − 14-s + 15-s − 16-s + 17-s − 18-s − 8·19-s − 20-s + 21-s + 4·22-s − 6·23-s + 3·24-s + 25-s + 27-s − 28-s − 6·29-s − 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s − 0.288·12-s − 0.267·14-s + 0.258·15-s − 1/4·16-s + 0.242·17-s − 0.235·18-s − 1.83·19-s − 0.223·20-s + 0.218·21-s + 0.852·22-s − 1.25·23-s + 0.612·24-s + 1/5·25-s + 0.192·27-s − 0.188·28-s − 1.11·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.10777869661756, −12.83550541164147, −12.38057855219446, −11.79104041206793, −10.89052562678825, −10.70667062743153, −10.47284572495408, −9.799858551716943, −9.493800387893523, −8.857869827522863, −8.653072214957393, −8.104912538688657, −7.725038444438117, −7.329461692341651, −6.794748551707451, −5.959162049753877, −5.598511892431251, −5.138845978181781, −4.451631907431155, −4.040091218264980, −3.617026948415958, −2.721199937149640, −2.134070200081646, −1.872568684497406, −1.166191504149526, 0, 0,
1.166191504149526, 1.872568684497406, 2.134070200081646, 2.721199937149640, 3.617026948415958, 4.040091218264980, 4.451631907431155, 5.138845978181781, 5.598511892431251, 5.959162049753877, 6.794748551707451, 7.329461692341651, 7.725038444438117, 8.104912538688657, 8.653072214957393, 8.857869827522863, 9.493800387893523, 9.799858551716943, 10.47284572495408, 10.70667062743153, 10.89052562678825, 11.79104041206793, 12.38057855219446, 12.83550541164147, 13.10777869661756
