| L(s) = 1 | − 9·3-s − 39.2·5-s − 42.0·7-s + 81·9-s − 168.·11-s − 876.·13-s + 353.·15-s + 1.45e3·17-s + 2.42e3·19-s + 378.·21-s − 529·23-s − 1.58e3·25-s − 729·27-s − 317.·29-s − 5.84e3·31-s + 1.51e3·33-s + 1.64e3·35-s − 1.46e4·37-s + 7.89e3·39-s − 1.98e4·41-s + 5.56e3·43-s − 3.17e3·45-s + 6.05e3·47-s − 1.50e4·49-s − 1.31e4·51-s − 1.07e4·53-s + 6.59e3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.701·5-s − 0.324·7-s + 0.333·9-s − 0.419·11-s − 1.43·13-s + 0.405·15-s + 1.22·17-s + 1.53·19-s + 0.187·21-s − 0.208·23-s − 0.507·25-s − 0.192·27-s − 0.0701·29-s − 1.09·31-s + 0.241·33-s + 0.227·35-s − 1.76·37-s + 0.830·39-s − 1.84·41-s + 0.458·43-s − 0.233·45-s + 0.399·47-s − 0.894·49-s − 0.706·51-s − 0.524·53-s + 0.294·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.5680306643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5680306643\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 9T \) |
| 23 | \( 1 + 529T \) |
| good | 5 | \( 1 + 39.2T + 3.12e3T^{2} \) |
| 7 | \( 1 + 42.0T + 1.68e4T^{2} \) |
| 11 | \( 1 + 168.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 876.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.45e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.42e3T + 2.47e6T^{2} \) |
| 29 | \( 1 + 317.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.84e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.46e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.98e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.56e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 6.05e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.07e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.35e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.83e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.70e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.22e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.59e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.00e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.86e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.88e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.00e5T + 8.58e9T^{2} \) |
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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
−9.351189981780410264845240296290, −8.016844107107819598728680296794, −7.48257586518476601223322848805, −6.78190226984167160160179791236, −5.41862514817343775183821897877, −5.13466833336789850655241290717, −3.80205306546685742572557330397, −3.03206207960229043807156160548, −1.64229959456085883250278850500, −0.32981143411936370742656374326,
0.32981143411936370742656374326, 1.64229959456085883250278850500, 3.03206207960229043807156160548, 3.80205306546685742572557330397, 5.13466833336789850655241290717, 5.41862514817343775183821897877, 6.78190226984167160160179791236, 7.48257586518476601223322848805, 8.016844107107819598728680296794, 9.351189981780410264845240296290
