| L(s) = 1 | − 5-s + 4·11-s − 13-s + 2·17-s + 4·19-s − 4·23-s + 25-s − 6·29-s − 4·31-s − 6·37-s + 2·41-s − 12·43-s − 8·47-s − 7·49-s + 14·53-s − 4·55-s + 12·59-s + 2·61-s + 65-s − 8·71-s − 2·73-s + 8·79-s + 12·83-s − 2·85-s − 6·89-s − 4·95-s − 10·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.20·11-s − 0.277·13-s + 0.485·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s − 1.11·29-s − 0.718·31-s − 0.986·37-s + 0.312·41-s − 1.82·43-s − 1.16·47-s − 49-s + 1.92·53-s − 0.539·55-s + 1.56·59-s + 0.256·61-s + 0.124·65-s − 0.949·71-s − 0.234·73-s + 0.900·79-s + 1.31·83-s − 0.216·85-s − 0.635·89-s − 0.410·95-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.785504925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.785504925\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.81391317905047, −14.46926492733788, −13.87512889256410, −13.30583996635737, −12.77359272748387, −12.07734687007834, −11.64574684576531, −11.50818406458104, −10.61744692717632, −10.01506158616489, −9.585411157634181, −9.026874663411260, −8.397813058299991, −7.903057245957798, −7.166877533771542, −6.881355527944435, −6.123732693721227, −5.429774963710095, −4.986430333642277, −4.068043156162820, −3.639583294038710, −3.145372135021737, −2.031291740462357, −1.488119630197809, −0.5014756483255785,
0.5014756483255785, 1.488119630197809, 2.031291740462357, 3.145372135021737, 3.639583294038710, 4.068043156162820, 4.986430333642277, 5.429774963710095, 6.123732693721227, 6.881355527944435, 7.166877533771542, 7.903057245957798, 8.397813058299991, 9.026874663411260, 9.585411157634181, 10.01506158616489, 10.61744692717632, 11.50818406458104, 11.64574684576531, 12.07734687007834, 12.77359272748387, 13.30583996635737, 13.87512889256410, 14.46926492733788, 14.81391317905047
