| L(s) = 1 | + 2.21·2-s − 3.10·4-s − 17.1·7-s − 24.5·8-s + 66.8·11-s − 72.7·13-s − 37.8·14-s − 29.5·16-s − 40.2·17-s + 38.4·19-s + 147.·22-s + 204.·23-s − 160.·26-s + 53.1·28-s − 21.6·29-s + 128.·31-s + 131.·32-s − 89.1·34-s − 19.4·37-s + 85.1·38-s + 270.·41-s − 242.·43-s − 207.·44-s + 452.·46-s + 307.·47-s − 50.4·49-s + 225.·52-s + ⋯ |
| L(s) = 1 | + 0.782·2-s − 0.388·4-s − 0.923·7-s − 1.08·8-s + 1.83·11-s − 1.55·13-s − 0.722·14-s − 0.461·16-s − 0.574·17-s + 0.464·19-s + 1.43·22-s + 1.85·23-s − 1.21·26-s + 0.358·28-s − 0.138·29-s + 0.743·31-s + 0.724·32-s − 0.449·34-s − 0.0862·37-s + 0.363·38-s + 1.02·41-s − 0.861·43-s − 0.711·44-s + 1.45·46-s + 0.954·47-s − 0.147·49-s + 0.602·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.072210907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.072210907\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 2.21T + 8T^{2} \) |
| 7 | \( 1 + 17.1T + 343T^{2} \) |
| 11 | \( 1 - 66.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 72.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 40.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 38.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 204.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 21.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 128.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 19.4T + 5.06e4T^{2} \) |
| 41 | \( 1 - 270.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 242.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 307.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 289.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 17.7T + 2.05e5T^{2} \) |
| 61 | \( 1 - 764.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 532.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 409.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 220.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.13e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 253.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.62e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 457.T + 9.12e5T^{2} \) |
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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
−9.718761770695042370823022760279, −9.443577991772599826118952398375, −8.601991501488461023571142924042, −7.04662050089853674182171648641, −6.57828996309685446187554956697, −5.42038387723303302553288337354, −4.51679512043230874224540587381, −3.63011578729043280126245587572, −2.64353702544237096647112489095, −0.74285780043883536650066637100,
0.74285780043883536650066637100, 2.64353702544237096647112489095, 3.63011578729043280126245587572, 4.51679512043230874224540587381, 5.42038387723303302553288337354, 6.57828996309685446187554956697, 7.04662050089853674182171648641, 8.601991501488461023571142924042, 9.443577991772599826118952398375, 9.718761770695042370823022760279
