| L(s) = 1 | + 4.55·2-s + 12.7·4-s + 17.8·5-s + 21.6·8-s + 81.4·10-s + 11.3·11-s + 13.0·13-s − 3.25·16-s + 53.2·17-s + 42.4·19-s + 228.·20-s + 51.9·22-s − 152.·23-s + 194.·25-s + 59.6·26-s − 186.·29-s + 157.·31-s − 188.·32-s + 242.·34-s + 3.74·37-s + 193.·38-s + 387.·40-s − 39.3·41-s + 429.·43-s + 145.·44-s − 692.·46-s + 21.1·47-s + ⋯ |
| L(s) = 1 | + 1.61·2-s + 1.59·4-s + 1.59·5-s + 0.958·8-s + 2.57·10-s + 0.312·11-s + 0.279·13-s − 0.0508·16-s + 0.759·17-s + 0.512·19-s + 2.54·20-s + 0.503·22-s − 1.37·23-s + 1.55·25-s + 0.450·26-s − 1.19·29-s + 0.914·31-s − 1.04·32-s + 1.22·34-s + 0.0166·37-s + 0.825·38-s + 1.53·40-s − 0.149·41-s + 1.52·43-s + 0.498·44-s − 2.22·46-s + 0.0657·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.407208894\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.407208894\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 4.55T + 8T^{2} \) |
| 5 | \( 1 - 17.8T + 125T^{2} \) |
| 11 | \( 1 - 11.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 13.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 53.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 42.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 152.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 186.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 157.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 3.74T + 5.06e4T^{2} \) |
| 41 | \( 1 + 39.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 429.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 21.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + 365.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 226.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 651.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 145.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 368.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 608.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 910.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 327.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 37.6T + 7.04e5T^{2} \) |
| 97 | \( 1 + 722.T + 9.12e5T^{2} \) |
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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
−10.88102843010757965665704512279, −9.900094430826861654359504103651, −9.140808012106877187060352705007, −7.62761357201801712765308720281, −6.32601169107515182862465857301, −5.90019721536507449179612574862, −5.05892880322225018591097400792, −3.86493389151354820441462091201, −2.69630191755323766432626524439, −1.60819519530947805331081525032,
1.60819519530947805331081525032, 2.69630191755323766432626524439, 3.86493389151354820441462091201, 5.05892880322225018591097400792, 5.90019721536507449179612574862, 6.32601169107515182862465857301, 7.62761357201801712765308720281, 9.140808012106877187060352705007, 9.900094430826861654359504103651, 10.88102843010757965665704512279
