| L(s) = 1 | + 8.82·2-s + 45.9·4-s + 87.3·5-s − 52.6·7-s + 123.·8-s + 771.·10-s + 273.·11-s + 908.·13-s − 465.·14-s − 382.·16-s − 501.·17-s − 136.·19-s + 4.01e3·20-s + 2.41e3·22-s + 529·23-s + 4.50e3·25-s + 8.02e3·26-s − 2.42e3·28-s − 1.70e3·29-s + 8.48e3·31-s − 7.32e3·32-s − 4.43e3·34-s − 4.60e3·35-s − 1.22e3·37-s − 1.20e3·38-s + 1.07e4·40-s − 6.33e3·41-s + ⋯ |
| L(s) = 1 | + 1.56·2-s + 1.43·4-s + 1.56·5-s − 0.406·7-s + 0.680·8-s + 2.43·10-s + 0.680·11-s + 1.49·13-s − 0.634·14-s − 0.373·16-s − 0.421·17-s − 0.0867·19-s + 2.24·20-s + 1.06·22-s + 0.208·23-s + 1.44·25-s + 2.32·26-s − 0.583·28-s − 0.375·29-s + 1.58·31-s − 1.26·32-s − 0.657·34-s − 0.635·35-s − 0.147·37-s − 0.135·38-s + 1.06·40-s − 0.588·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(6.429287125\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.429287125\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - 529T \) |
| good | 2 | \( 1 - 8.82T + 32T^{2} \) |
| 5 | \( 1 - 87.3T + 3.12e3T^{2} \) |
| 7 | \( 1 + 52.6T + 1.68e4T^{2} \) |
| 11 | \( 1 - 273.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 908.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 501.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 136.T + 2.47e6T^{2} \) |
| 29 | \( 1 + 1.70e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.48e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.22e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.33e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.24e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.86e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.84e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.59e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.32e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.91e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.18e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.09e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.87e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.65e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.03e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.56e5T + 8.58e9T^{2} \) |
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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
−11.77244428314691651486324074964, −10.77559217363953498546682236502, −9.646655446398204221381480677852, −8.680609884148869747422541605572, −6.60363749350828701747684324116, −6.23506287694751466835921795408, −5.25766839286698289779812127588, −3.98595862302851482825341889640, −2.79754852177684371846487394284, −1.50262370166773459278501262699,
1.50262370166773459278501262699, 2.79754852177684371846487394284, 3.98595862302851482825341889640, 5.25766839286698289779812127588, 6.23506287694751466835921795408, 6.60363749350828701747684324116, 8.680609884148869747422541605572, 9.646655446398204221381480677852, 10.77559217363953498546682236502, 11.77244428314691651486324074964
