| L(s) = 1 | − 5-s − 1.43·7-s − 1.35·11-s − 5.52·13-s − 4.82·17-s + 0.648·19-s − 8.90·23-s + 25-s + 7.17·29-s + 4.64·31-s + 1.43·35-s + 1.35·37-s − 0.351·41-s − 4.82·43-s − 9.49·47-s − 4.93·49-s + 8.17·53-s + 1.35·55-s + 1.46·59-s + 6.69·61-s + 5.52·65-s − 12.4·67-s + 2.22·71-s − 4.34·73-s + 1.94·77-s + 13.0·79-s − 5.26·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.543·7-s − 0.407·11-s − 1.53·13-s − 1.16·17-s + 0.148·19-s − 1.85·23-s + 0.200·25-s + 1.33·29-s + 0.834·31-s + 0.243·35-s + 0.222·37-s − 0.0549·41-s − 0.735·43-s − 1.38·47-s − 0.704·49-s + 1.12·53-s + 0.182·55-s + 0.191·59-s + 0.857·61-s + 0.685·65-s − 1.51·67-s + 0.264·71-s − 0.508·73-s + 0.221·77-s + 1.46·79-s − 0.577·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7493599205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7493599205\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 7 | \( 1 + 1.43T + 7T^{2} \) |
| 11 | \( 1 + 1.35T + 11T^{2} \) |
| 13 | \( 1 + 5.52T + 13T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 19 | \( 1 - 0.648T + 19T^{2} \) |
| 23 | \( 1 + 8.90T + 23T^{2} \) |
| 29 | \( 1 - 7.17T + 29T^{2} \) |
| 31 | \( 1 - 4.64T + 31T^{2} \) |
| 37 | \( 1 - 1.35T + 37T^{2} \) |
| 41 | \( 1 + 0.351T + 41T^{2} \) |
| 43 | \( 1 + 4.82T + 43T^{2} \) |
| 47 | \( 1 + 9.49T + 47T^{2} \) |
| 53 | \( 1 - 8.17T + 53T^{2} \) |
| 59 | \( 1 - 1.46T + 59T^{2} \) |
| 61 | \( 1 - 6.69T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 - 2.22T + 71T^{2} \) |
| 73 | \( 1 + 4.34T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 + 5.26T + 83T^{2} \) |
| 89 | \( 1 - 11T + 89T^{2} \) |
| 97 | \( 1 - 17.5T + 97T^{2} \) |
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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
−8.023503997128493304099233956862, −7.32536389848072229252557509874, −6.60728077523054447225699803564, −6.05963657036066672633582773840, −4.90848065314951785125087152897, −4.56975264913423021442671271604, −3.58562439507220593584551691652, −2.70353089847038432414559472636, −2.02931640904530562770810775721, −0.41907193708444601593264629393,
0.41907193708444601593264629393, 2.02931640904530562770810775721, 2.70353089847038432414559472636, 3.58562439507220593584551691652, 4.56975264913423021442671271604, 4.90848065314951785125087152897, 6.05963657036066672633582773840, 6.60728077523054447225699803564, 7.32536389848072229252557509874, 8.023503997128493304099233956862
