| L(s) = 1 | − 2.90·3-s − 5-s − 1.52·7-s + 5.42·9-s + 0.903·11-s + 2.42·13-s + 2.90·15-s + 2.28·17-s − 0.474·19-s + 4.42·21-s − 2.90·23-s + 25-s − 7.05·27-s + 29-s + 5.33·31-s − 2.62·33-s + 1.52·35-s − 3.52·37-s − 7.05·39-s + 4.62·41-s − 12.7·43-s − 5.42·45-s − 1.65·47-s − 4.67·49-s − 6.62·51-s − 2.13·53-s − 0.903·55-s + ⋯ |
| L(s) = 1 | − 1.67·3-s − 0.447·5-s − 0.576·7-s + 1.80·9-s + 0.272·11-s + 0.673·13-s + 0.749·15-s + 0.553·17-s − 0.108·19-s + 0.966·21-s − 0.605·23-s + 0.200·25-s − 1.35·27-s + 0.185·29-s + 0.957·31-s − 0.456·33-s + 0.257·35-s − 0.579·37-s − 1.12·39-s + 0.721·41-s − 1.93·43-s − 0.809·45-s − 0.241·47-s − 0.667·49-s − 0.927·51-s − 0.293·53-s − 0.121·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 2.90T + 3T^{2} \) |
| 7 | \( 1 + 1.52T + 7T^{2} \) |
| 11 | \( 1 - 0.903T + 11T^{2} \) |
| 13 | \( 1 - 2.42T + 13T^{2} \) |
| 17 | \( 1 - 2.28T + 17T^{2} \) |
| 19 | \( 1 + 0.474T + 19T^{2} \) |
| 23 | \( 1 + 2.90T + 23T^{2} \) |
| 31 | \( 1 - 5.33T + 31T^{2} \) |
| 37 | \( 1 + 3.52T + 37T^{2} \) |
| 41 | \( 1 - 4.62T + 41T^{2} \) |
| 43 | \( 1 + 12.7T + 43T^{2} \) |
| 47 | \( 1 + 1.65T + 47T^{2} \) |
| 53 | \( 1 + 2.13T + 53T^{2} \) |
| 59 | \( 1 + 7.18T + 59T^{2} \) |
| 61 | \( 1 + 7.67T + 61T^{2} \) |
| 67 | \( 1 + 2.04T + 67T^{2} \) |
| 71 | \( 1 - 3.18T + 71T^{2} \) |
| 73 | \( 1 + 8.90T + 73T^{2} \) |
| 79 | \( 1 - 5.19T + 79T^{2} \) |
| 83 | \( 1 + 15.6T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 - 3.39T + 97T^{2} \) |
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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
−9.679442549492682241885612849173, −8.520639671294869211340759654291, −7.54798663079659319533512304545, −6.52323079437938361719919624975, −6.16983810774882504032387948551, −5.16991604627669649020987698237, −4.31193595951098153514019315954, −3.27566720763600159230373872435, −1.34860436452899189187683807873, 0,
1.34860436452899189187683807873, 3.27566720763600159230373872435, 4.31193595951098153514019315954, 5.16991604627669649020987698237, 6.16983810774882504032387948551, 6.52323079437938361719919624975, 7.54798663079659319533512304545, 8.520639671294869211340759654291, 9.679442549492682241885612849173
