| L(s) = 1 | − 4.54·2-s + 12.6·4-s − 6.53·5-s + 7·7-s − 21.0·8-s + 29.6·10-s − 11·11-s + 71.3·13-s − 31.8·14-s − 5.29·16-s − 2.45·17-s + 80.0·19-s − 82.6·20-s + 49.9·22-s − 61.8·23-s − 82.2·25-s − 324.·26-s + 88.5·28-s + 156.·29-s + 77.7·31-s + 192.·32-s + 11.1·34-s − 45.7·35-s + 84.7·37-s − 363.·38-s + 137.·40-s − 28.8·41-s + ⋯ |
| L(s) = 1 | − 1.60·2-s + 1.58·4-s − 0.584·5-s + 0.377·7-s − 0.932·8-s + 0.939·10-s − 0.301·11-s + 1.52·13-s − 0.607·14-s − 0.0827·16-s − 0.0349·17-s + 0.966·19-s − 0.923·20-s + 0.484·22-s − 0.560·23-s − 0.658·25-s − 2.44·26-s + 0.597·28-s + 1.00·29-s + 0.450·31-s + 1.06·32-s + 0.0561·34-s − 0.220·35-s + 0.376·37-s − 1.55·38-s + 0.545·40-s − 0.109·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8019263143\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8019263143\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 4.54T + 8T^{2} \) |
| 5 | \( 1 + 6.53T + 125T^{2} \) |
| 13 | \( 1 - 71.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 2.45T + 4.91e3T^{2} \) |
| 19 | \( 1 - 80.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 61.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 156.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 77.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 84.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 28.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 352.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 256.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 492.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 3.12T + 2.05e5T^{2} \) |
| 61 | \( 1 + 159.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 521.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 885.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 375.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.34e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.37e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 111.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 472.T + 9.12e5T^{2} \) |
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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
−10.00245988880718076219434094945, −9.115583821861748867683411508412, −8.221461076506843792233610510284, −7.919897551605345169482123350504, −6.87619158569995011516680490295, −5.89344698451128017408896325060, −4.46713710308949712510447831323, −3.19799877119024357063671321045, −1.72084336848014269129520919586, −0.68478620519020112135929047380,
0.68478620519020112135929047380, 1.72084336848014269129520919586, 3.19799877119024357063671321045, 4.46713710308949712510447831323, 5.89344698451128017408896325060, 6.87619158569995011516680490295, 7.919897551605345169482123350504, 8.221461076506843792233610510284, 9.115583821861748867683411508412, 10.00245988880718076219434094945
