| L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 5-s − 2·6-s − 5·7-s − 2·9-s + 2·10-s + 3·11-s + 2·12-s − 2·13-s + 10·14-s − 15-s − 4·16-s − 4·17-s + 4·18-s − 4·19-s − 2·20-s − 5·21-s − 6·22-s − 2·23-s + 25-s + 4·26-s − 5·27-s − 10·28-s + 2·29-s + 2·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 0.447·5-s − 0.816·6-s − 1.88·7-s − 2/3·9-s + 0.632·10-s + 0.904·11-s + 0.577·12-s − 0.554·13-s + 2.67·14-s − 0.258·15-s − 16-s − 0.970·17-s + 0.942·18-s − 0.917·19-s − 0.447·20-s − 1.09·21-s − 1.27·22-s − 0.417·23-s + 1/5·25-s + 0.784·26-s − 0.962·27-s − 1.88·28-s + 0.371·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−11.96118951087323707567154728209, −10.80170433061064572152121626853, −9.758331603941689019220020572145, −9.108040044433494028384347405100, −8.403685986159716877085877336376, −7.11635102647622048765197516830, −6.31090460671582344240143237850, −3.99459552465236235066187455838, −2.56821315774302943429182728141, 0,
2.56821315774302943429182728141, 3.99459552465236235066187455838, 6.31090460671582344240143237850, 7.11635102647622048765197516830, 8.403685986159716877085877336376, 9.108040044433494028384347405100, 9.758331603941689019220020572145, 10.80170433061064572152121626853, 11.96118951087323707567154728209
