L(s) = 1 | + 5-s − 11-s + 7·13-s − 2·17-s + 2·23-s + 25-s − 2·29-s + 3·31-s + 12·37-s − 6·41-s − 43-s − 10·47-s − 55-s + 7·59-s + 7·65-s + 4·67-s + 9·71-s + 9·73-s + 6·79-s + 11·83-s − 2·85-s − 7·89-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.301·11-s + 1.94·13-s − 0.485·17-s + 0.417·23-s + 1/5·25-s − 0.371·29-s + 0.538·31-s + 1.97·37-s − 0.937·41-s − 0.152·43-s − 1.45·47-s − 0.134·55-s + 0.911·59-s + 0.868·65-s + 0.488·67-s + 1.06·71-s + 1.05·73-s + 0.675·79-s + 1.20·83-s − 0.216·85-s − 0.741·89-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.685696820\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.685696820\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 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
−12.63446175364391, −11.94578635073520, −11.37240227792103, −11.10235254190546, −10.83091282716920, −10.19065243186169, −9.662086555920499, −9.420879530821638, −8.737096759490587, −8.365525115993609, −8.065906249601581, −7.478329399874357, −6.685799135541005, −6.476294460680163, −6.125547579580267, −5.383103910264545, −5.153625369655832, −4.413556297219580, −3.921296818091021, −3.449768917490382, −2.869691654994540, −2.294459663190067, −1.671329864641733, −1.106966812006244, −0.5325756948519706,
0.5325756948519706, 1.106966812006244, 1.671329864641733, 2.294459663190067, 2.869691654994540, 3.449768917490382, 3.921296818091021, 4.413556297219580, 5.153625369655832, 5.383103910264545, 6.125547579580267, 6.476294460680163, 6.685799135541005, 7.478329399874357, 8.065906249601581, 8.365525115993609, 8.737096759490587, 9.420879530821638, 9.662086555920499, 10.19065243186169, 10.83091282716920, 11.10235254190546, 11.37240227792103, 11.94578635073520, 12.63446175364391