| L(s) = 1 | − 5-s − 3·7-s + 5·11-s − 5·13-s − 2·17-s − 4·19-s − 23-s − 4·25-s − 9·29-s − 31-s + 3·35-s − 6·37-s + 3·41-s + 43-s − 3·47-s + 2·49-s + 2·53-s − 5·55-s + 11·59-s + 7·61-s + 5·65-s − 67-s + 4·71-s − 2·73-s − 15·77-s + 79-s + 83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.13·7-s + 1.50·11-s − 1.38·13-s − 0.485·17-s − 0.917·19-s − 0.208·23-s − 4/5·25-s − 1.67·29-s − 0.179·31-s + 0.507·35-s − 0.986·37-s + 0.468·41-s + 0.152·43-s − 0.437·47-s + 2/7·49-s + 0.274·53-s − 0.674·55-s + 1.43·59-s + 0.896·61-s + 0.620·65-s − 0.122·67-s + 0.474·71-s − 0.234·73-s − 1.70·77-s + 0.112·79-s + 0.109·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−9.872166086054390182131680931495, −9.425062765356892372201588625238, −8.488693062062992053262410459368, −7.24964163655286011443852633632, −6.70196192600962287834875283080, −5.67914881339550181972392218784, −4.28403219298754109102003210043, −3.56271769221553523884362414547, −2.12247216163999118504559644758, 0,
2.12247216163999118504559644758, 3.56271769221553523884362414547, 4.28403219298754109102003210043, 5.67914881339550181972392218784, 6.70196192600962287834875283080, 7.24964163655286011443852633632, 8.488693062062992053262410459368, 9.425062765356892372201588625238, 9.872166086054390182131680931495
