| L(s) = 1 | − 9.12·2-s + 51.1·4-s − 25·5-s + 202.·7-s − 175.·8-s + 228.·10-s − 121·11-s − 622.·13-s − 1.84e3·14-s − 41.4·16-s + 1.58e3·17-s + 1.44e3·19-s − 1.27e3·20-s + 1.10e3·22-s + 1.55e3·23-s + 625·25-s + 5.68e3·26-s + 1.03e4·28-s − 4.45e3·29-s + 5.06e3·31-s + 5.97e3·32-s − 1.44e4·34-s − 5.06e3·35-s + 1.16e4·37-s − 1.31e4·38-s + 4.37e3·40-s + 4.84e3·41-s + ⋯ |
| L(s) = 1 | − 1.61·2-s + 1.59·4-s − 0.447·5-s + 1.56·7-s − 0.967·8-s + 0.721·10-s − 0.301·11-s − 1.02·13-s − 2.51·14-s − 0.0404·16-s + 1.33·17-s + 0.917·19-s − 0.715·20-s + 0.486·22-s + 0.614·23-s + 0.200·25-s + 1.64·26-s + 2.49·28-s − 0.982·29-s + 0.946·31-s + 1.03·32-s − 2.15·34-s − 0.698·35-s + 1.40·37-s − 1.47·38-s + 0.432·40-s + 0.449·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.047127195\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.047127195\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 25T \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 + 9.12T + 32T^{2} \) |
| 7 | \( 1 - 202.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 622.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.58e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.44e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.55e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.45e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.06e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.16e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 4.84e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.92e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.33e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 4.24e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.32e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.99e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 7.11e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.21e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.91e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.52e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.18e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.41e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.55e4T + 8.58e9T^{2} \) |
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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
−10.01533092784295922374729912068, −9.273734654455984778868168503003, −8.165267862833789027622894691208, −7.77053829711525987871123146117, −7.13210497203354516986812708269, −5.49144176698468777144431223837, −4.54284713898804226994618388559, −2.82530711685101660172679659366, −1.59149503666426475324186334083, −0.70521059955991231548036340780,
0.70521059955991231548036340780, 1.59149503666426475324186334083, 2.82530711685101660172679659366, 4.54284713898804226994618388559, 5.49144176698468777144431223837, 7.13210497203354516986812708269, 7.77053829711525987871123146117, 8.165267862833789027622894691208, 9.273734654455984778868168503003, 10.01533092784295922374729912068
