| L(s) = 1 | − 2-s − 4-s + 5-s − 3·7-s + 3·8-s − 10-s + 2·11-s − 2·13-s + 3·14-s − 16-s − 4·17-s − 8·19-s − 20-s − 2·22-s − 3·23-s + 25-s + 2·26-s + 3·28-s + 29-s − 5·32-s + 4·34-s − 3·35-s − 4·37-s + 8·38-s + 3·40-s − 5·41-s − 8·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s − 1.13·7-s + 1.06·8-s − 0.316·10-s + 0.603·11-s − 0.554·13-s + 0.801·14-s − 1/4·16-s − 0.970·17-s − 1.83·19-s − 0.223·20-s − 0.426·22-s − 0.625·23-s + 1/5·25-s + 0.392·26-s + 0.566·28-s + 0.185·29-s − 0.883·32-s + 0.685·34-s − 0.507·35-s − 0.657·37-s + 1.29·38-s + 0.474·40-s − 0.780·41-s − 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 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
−10.32043257877866718022672062677, −9.946177052534403132870451803520, −8.964305577132556078601253128051, −8.406907006225460062267410041104, −6.95907981187563203473208917697, −6.29557968587596560657178728455, −4.84628635731234228377894729907, −3.75836640301739809806939141128, −2.06014522909482997436709887093, 0,
2.06014522909482997436709887093, 3.75836640301739809806939141128, 4.84628635731234228377894729907, 6.29557968587596560657178728455, 6.95907981187563203473208917697, 8.406907006225460062267410041104, 8.964305577132556078601253128051, 9.946177052534403132870451803520, 10.32043257877866718022672062677
