| L(s) = 1 | − 3·3-s − 15.3·5-s + 7·7-s + 9·9-s − 5.35·11-s − 11.2·13-s + 46.0·15-s + 94.0·17-s + 20·19-s − 21·21-s − 102.·23-s + 110.·25-s − 27·27-s − 102·29-s − 341.·31-s + 16.0·33-s − 107.·35-s − 288.·37-s + 33.8·39-s − 252.·41-s − 145.·43-s − 138.·45-s + 573.·47-s + 49·49-s − 282.·51-s + 234.·53-s + 82.2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.37·5-s + 0.377·7-s + 0.333·9-s − 0.146·11-s − 0.240·13-s + 0.793·15-s + 1.34·17-s + 0.241·19-s − 0.218·21-s − 0.931·23-s + 0.886·25-s − 0.192·27-s − 0.653·29-s − 1.97·31-s + 0.0847·33-s − 0.519·35-s − 1.28·37-s + 0.139·39-s − 0.963·41-s − 0.516·43-s − 0.457·45-s + 1.78·47-s + 0.142·49-s − 0.774·51-s + 0.607·53-s + 0.201·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8258424305\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8258424305\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 - 7T \) |
| good | 5 | \( 1 + 15.3T + 125T^{2} \) |
| 11 | \( 1 + 5.35T + 1.33e3T^{2} \) |
| 13 | \( 1 + 11.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 94.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 20T + 6.85e3T^{2} \) |
| 23 | \( 1 + 102.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 102T + 2.43e4T^{2} \) |
| 31 | \( 1 + 341.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 288.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 252.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 145.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 573.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 234.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 151.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 243.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 142.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 65.0T + 3.57e5T^{2} \) |
| 73 | \( 1 - 380.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 830.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 469.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.55e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.08e3T + 9.12e5T^{2} \) |
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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
−9.210949334217546792514199524444, −8.226566167661002014885380645478, −7.56701942131281943477060632010, −7.05299174035324700957338075233, −5.72372483680340028181148966939, −5.11265456856651801589645306924, −3.99741477738741610552388225265, −3.41151339734414707905532399432, −1.79952437172019671701933941579, −0.46663864824742395026595539611,
0.46663864824742395026595539611, 1.79952437172019671701933941579, 3.41151339734414707905532399432, 3.99741477738741610552388225265, 5.11265456856651801589645306924, 5.72372483680340028181148966939, 7.05299174035324700957338075233, 7.56701942131281943477060632010, 8.226566167661002014885380645478, 9.210949334217546792514199524444
