| L(s) = 1 | − 3.11·3-s + 3.47·5-s − 7-s + 6.70·9-s + 4.22·11-s + 1.28·13-s − 10.8·15-s − 17-s + 2.94·19-s + 3.11·21-s + 0.715·23-s + 7.06·25-s − 11.5·27-s + 6.94·29-s − 8.98·31-s − 13.1·33-s − 3.47·35-s + 10.9·37-s − 4.00·39-s + 2.06·41-s − 0.399·43-s + 23.2·45-s + 12.4·47-s + 49-s + 3.11·51-s − 6.64·53-s + 14.6·55-s + ⋯ |
| L(s) = 1 | − 1.79·3-s + 1.55·5-s − 0.377·7-s + 2.23·9-s + 1.27·11-s + 0.356·13-s − 2.79·15-s − 0.242·17-s + 0.675·19-s + 0.679·21-s + 0.149·23-s + 1.41·25-s − 2.21·27-s + 1.28·29-s − 1.61·31-s − 2.29·33-s − 0.587·35-s + 1.79·37-s − 0.640·39-s + 0.321·41-s − 0.0608·43-s + 3.46·45-s + 1.81·47-s + 0.142·49-s + 0.436·51-s − 0.913·53-s + 1.98·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.796162440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.796162440\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 3.11T + 3T^{2} \) |
| 5 | \( 1 - 3.47T + 5T^{2} \) |
| 11 | \( 1 - 4.22T + 11T^{2} \) |
| 13 | \( 1 - 1.28T + 13T^{2} \) |
| 19 | \( 1 - 2.94T + 19T^{2} \) |
| 23 | \( 1 - 0.715T + 23T^{2} \) |
| 29 | \( 1 - 6.94T + 29T^{2} \) |
| 31 | \( 1 + 8.98T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 - 2.06T + 41T^{2} \) |
| 43 | \( 1 + 0.399T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 + 6.64T + 53T^{2} \) |
| 59 | \( 1 + 1.43T + 59T^{2} \) |
| 61 | \( 1 - 6.88T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 + 14.5T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 1.05T + 83T^{2} \) |
| 89 | \( 1 - 4.60T + 89T^{2} \) |
| 97 | \( 1 - 10.2T + 97T^{2} \) |
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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
−7.55154814530220061600860057977, −6.71861174168812901647576859849, −6.41822014702298546737006540583, −5.80769598634998352812640450247, −5.37360625481194901224286903815, −4.53831378585146257757118163710, −3.76515129597664665357214246115, −2.47202387558783589583912477962, −1.42793343644474169913991114012, −0.835242867009325714520939706967,
0.835242867009325714520939706967, 1.42793343644474169913991114012, 2.47202387558783589583912477962, 3.76515129597664665357214246115, 4.53831378585146257757118163710, 5.37360625481194901224286903815, 5.80769598634998352812640450247, 6.41822014702298546737006540583, 6.71861174168812901647576859849, 7.55154814530220061600860057977
