| L(s) = 1 | + 3.63·2-s + 5.18·4-s − 21.1·5-s + 7·7-s − 10.2·8-s − 76.6·10-s − 11·11-s + 87.3·13-s + 25.4·14-s − 78.6·16-s + 28.2·17-s − 97.9·19-s − 109.·20-s − 39.9·22-s + 112.·23-s + 320.·25-s + 317.·26-s + 36.2·28-s − 14.9·29-s + 138.·31-s − 203.·32-s + 102.·34-s − 147.·35-s + 206.·37-s − 355.·38-s + 215.·40-s − 321.·41-s + ⋯ |
| L(s) = 1 | + 1.28·2-s + 0.647·4-s − 1.88·5-s + 0.377·7-s − 0.452·8-s − 2.42·10-s − 0.301·11-s + 1.86·13-s + 0.485·14-s − 1.22·16-s + 0.403·17-s − 1.18·19-s − 1.22·20-s − 0.387·22-s + 1.01·23-s + 2.56·25-s + 2.39·26-s + 0.244·28-s − 0.0955·29-s + 0.802·31-s − 1.12·32-s + 0.517·34-s − 0.713·35-s + 0.919·37-s − 1.51·38-s + 0.853·40-s − 1.22·41-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
| \(L(2)\) |
\(\approx\) |
\(2.622370317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.622370317\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 3.63T + 8T^{2} \) |
| 5 | \( 1 + 21.1T + 125T^{2} \) |
| 13 | \( 1 - 87.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 28.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 97.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 112.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 14.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 138.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 206.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 321.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 285.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 303.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 554.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 693.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 156.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 584.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 363.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 747.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 419.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.17e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 397.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.33e3T + 9.12e5T^{2} \) |
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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
−10.53546410758396223170220680306, −8.703004181212584552021084316538, −8.443741241858736150161418802768, −7.30874993226106632022443840921, −6.38434472266800392644747030944, −5.30991150451472222951706877945, −4.23450970235591637826271717707, −3.86082219982395648308233352338, −2.84676433370662967029939020265, −0.77938499790889467037464318533,
0.77938499790889467037464318533, 2.84676433370662967029939020265, 3.86082219982395648308233352338, 4.23450970235591637826271717707, 5.30991150451472222951706877945, 6.38434472266800392644747030944, 7.30874993226106632022443840921, 8.443741241858736150161418802768, 8.703004181212584552021084316538, 10.53546410758396223170220680306
