| L(s) = 1 | − 0.732·2-s − 1.46·4-s + 2.73·5-s − 0.267·7-s + 2.53·8-s − 2·10-s + 11-s − 3.73·13-s + 0.196·14-s + 1.07·16-s + 6.19·17-s + 5.19·19-s − 4·20-s − 0.732·22-s + 8·23-s + 2.46·25-s + 2.73·26-s + 0.392·28-s − 4.73·29-s + 7.46·31-s − 5.85·32-s − 4.53·34-s − 0.732·35-s + 0.464·37-s − 3.80·38-s + 6.92·40-s − 5.46·41-s + ⋯ |
| L(s) = 1 | − 0.517·2-s − 0.732·4-s + 1.22·5-s − 0.101·7-s + 0.896·8-s − 0.632·10-s + 0.301·11-s − 1.03·13-s + 0.0524·14-s + 0.267·16-s + 1.50·17-s + 1.19·19-s − 0.894·20-s − 0.156·22-s + 1.66·23-s + 0.492·25-s + 0.535·26-s + 0.0741·28-s − 0.878·29-s + 1.34·31-s − 1.03·32-s − 0.777·34-s − 0.123·35-s + 0.0762·37-s − 0.617·38-s + 1.09·40-s − 0.853·41-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
| \(L(1)\) |
\(\approx\) |
\(1.060804225\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.060804225\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.732T + 2T^{2} \) |
| 5 | \( 1 - 2.73T + 5T^{2} \) |
| 7 | \( 1 + 0.267T + 7T^{2} \) |
| 13 | \( 1 + 3.73T + 13T^{2} \) |
| 17 | \( 1 - 6.19T + 17T^{2} \) |
| 19 | \( 1 - 5.19T + 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 + 4.73T + 29T^{2} \) |
| 31 | \( 1 - 7.46T + 31T^{2} \) |
| 37 | \( 1 - 0.464T + 37T^{2} \) |
| 41 | \( 1 + 5.46T + 41T^{2} \) |
| 43 | \( 1 + 3.46T + 43T^{2} \) |
| 47 | \( 1 + 0.196T + 47T^{2} \) |
| 53 | \( 1 - 1.26T + 53T^{2} \) |
| 59 | \( 1 + 0.196T + 59T^{2} \) |
| 61 | \( 1 + 7.73T + 61T^{2} \) |
| 67 | \( 1 + 7.92T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 + 7.19T + 73T^{2} \) |
| 79 | \( 1 + 2.26T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 17.6T + 89T^{2} \) |
| 97 | \( 1 - 15.3T + 97T^{2} \) |
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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
−11.77323412864000495240633816313, −10.39027671878994410004996861210, −9.689199696821821714230887196757, −9.295086249284438342105147771882, −8.024535477872286062261997860666, −7.02965517685962152420233710707, −5.60423020776044057136458859251, −4.88165301866530600637819555118, −3.13155787621081297820192150926, −1.32846942265979941203930312232,
1.32846942265979941203930312232, 3.13155787621081297820192150926, 4.88165301866530600637819555118, 5.60423020776044057136458859251, 7.02965517685962152420233710707, 8.024535477872286062261997860666, 9.295086249284438342105147771882, 9.689199696821821714230887196757, 10.39027671878994410004996861210, 11.77323412864000495240633816313
