| L(s) = 1 | + 2·2-s + 3-s + 2·4-s − 5-s + 2·6-s + 2·7-s + 9-s − 2·10-s − 11-s + 2·12-s − 13-s + 4·14-s − 15-s − 4·16-s − 4·17-s + 2·18-s − 6·19-s − 2·20-s + 2·21-s − 2·22-s − 6·23-s + 25-s − 2·26-s + 27-s + 4·28-s − 4·29-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 + 0.755·7-s + 1/3·9-s − 0.632·10-s − 0.301·11-s + 0.577·12-s − 0.277·13-s + 1.06·14-s − 0.258·15-s − 16-s − 0.970·17-s + 0.471·18-s − 1.37·19-s − 0.447·20-s + 0.436·21-s − 0.426·22-s − 1.25·23-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.755·28-s − 0.742·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.63624796455271497038986254722, −6.87794291995185622970121515143, −6.21783146575167091851519070713, −5.32244552148415198725030173280, −4.68178518117835223397180910712, −4.06962236451793862215646272971, −3.53315244344984619909485359094, −2.39536147801506979443252531730, −1.95211629619785178674943996841, 0,
1.95211629619785178674943996841, 2.39536147801506979443252531730, 3.53315244344984619909485359094, 4.06962236451793862215646272971, 4.68178518117835223397180910712, 5.32244552148415198725030173280, 6.21783146575167091851519070713, 6.87794291995185622970121515143, 7.63624796455271497038986254722
