L(s) = 1 | − 2.73·2-s + 5.46·4-s + 0.732·5-s − 3.73·7-s − 9.46·8-s − 2·10-s − 11-s − 0.267·13-s + 10.1·14-s + 14.9·16-s + 4.19·17-s − 5.19·19-s + 4·20-s + 2.73·22-s − 8·23-s − 4.46·25-s + 0.732·26-s − 20.3·28-s + 1.26·29-s + 0.535·31-s − 21.8·32-s − 11.4·34-s − 2.73·35-s − 6.46·37-s + 14.1·38-s − 6.92·40-s − 1.46·41-s + ⋯ |
L(s) = 1 | − 1.93·2-s + 2.73·4-s + 0.327·5-s − 1.41·7-s − 3.34·8-s − 0.632·10-s − 0.301·11-s − 0.0743·13-s + 2.72·14-s + 3.73·16-s + 1.01·17-s − 1.19·19-s + 0.894·20-s + 0.582·22-s − 1.66·23-s − 0.892·25-s + 0.143·26-s − 3.85·28-s + 0.235·29-s + 0.0962·31-s − 3.86·32-s − 1.96·34-s − 0.461·35-s − 1.06·37-s + 2.30·38-s − 1.09·40-s − 0.228·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + 2.73T + 2T^{2} \) |
| 5 | \( 1 - 0.732T + 5T^{2} \) |
| 7 | \( 1 + 3.73T + 7T^{2} \) |
| 13 | \( 1 + 0.267T + 13T^{2} \) |
| 17 | \( 1 - 4.19T + 17T^{2} \) |
| 19 | \( 1 + 5.19T + 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 - 1.26T + 29T^{2} \) |
| 31 | \( 1 - 0.535T + 31T^{2} \) |
| 37 | \( 1 + 6.46T + 37T^{2} \) |
| 41 | \( 1 + 1.46T + 41T^{2} \) |
| 43 | \( 1 - 3.46T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 + 4.73T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 + 4.26T + 61T^{2} \) |
| 67 | \( 1 - 5.92T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 - 3.19T + 73T^{2} \) |
| 79 | \( 1 + 5.73T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 0.339T + 89T^{2} \) |
| 97 | \( 1 + 5.39T + 97T^{2} \) |
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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
−10.82064368787251089401295184476, −9.922946070683342917394260083375, −9.699982788297855978122388060119, −8.511700238161327440662268732347, −7.69953954818979481840226331198, −6.55397274843630290556864924370, −5.94520275907457442358055304251, −3.35149336968103963333524756878, −2.02172474427898299129359430859, 0,
2.02172474427898299129359430859, 3.35149336968103963333524756878, 5.94520275907457442358055304251, 6.55397274843630290556864924370, 7.69953954818979481840226331198, 8.511700238161327440662268732347, 9.699982788297855978122388060119, 9.922946070683342917394260083375, 10.82064368787251089401295184476