| L(s) = 1 | − 2-s + 3-s − 4-s − 5-s − 6-s − 2·7-s + 3·8-s + 9-s + 10-s − 6·11-s − 12-s + 2·14-s − 15-s − 16-s − 6·17-s − 18-s + 19-s + 20-s − 2·21-s + 6·22-s − 8·23-s + 3·24-s + 25-s + 27-s + 2·28-s + 4·29-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 − 0.755·7-s + 1.06·8-s + 1/3·9-s + 0.316·10-s − 1.80·11-s − 0.288·12-s + 0.534·14-s − 0.258·15-s − 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.229·19-s + 0.223·20-s − 0.436·21-s + 1.27·22-s − 1.66·23-s + 0.612·24-s + 1/5·25-s + 0.192·27-s + 0.377·28-s + 0.742·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−11.04359797233522452957058795898, −10.14916660629762825787921129871, −9.508739877617597277399182494308, −8.311172749518175221259006108214, −7.944956574862892187546933534941, −6.69343741358824607018216997419, −5.08269165937541766478904797647, −3.93065235513561826916389859535, −2.45875404104195454497297562929, 0,
2.45875404104195454497297562929, 3.93065235513561826916389859535, 5.08269165937541766478904797647, 6.69343741358824607018216997419, 7.944956574862892187546933534941, 8.311172749518175221259006108214, 9.508739877617597277399182494308, 10.14916660629762825787921129871, 11.04359797233522452957058795898
