| L(s) = 1 | − 964. i·5-s + (2.36e3 + 397. i)7-s − 1.18e4·11-s − 2.88e4i·13-s − 4.60e4i·17-s + 1.94e4i·19-s + 1.31e5·23-s − 5.38e5·25-s − 1.22e6·29-s − 1.08e6i·31-s + (3.82e5 − 2.28e6i)35-s + 2.48e6·37-s − 2.64e4i·41-s − 4.08e6·43-s − 6.95e6i·47-s + ⋯ |
| L(s) = 1 | − 1.54i·5-s + (0.986 + 0.165i)7-s − 0.809·11-s − 1.01i·13-s − 0.550i·17-s + 0.149i·19-s + 0.470·23-s − 1.37·25-s − 1.73·29-s − 1.17i·31-s + (0.255 − 1.52i)35-s + 1.32·37-s − 0.00936i·41-s − 1.19·43-s − 1.42i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.165i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.986 - 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.181197370\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.181197370\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.36e3 - 397. i)T \) |
| good | 5 | \( 1 + 964. iT - 3.90e5T^{2} \) |
| 11 | \( 1 + 1.18e4T + 2.14e8T^{2} \) |
| 13 | \( 1 + 2.88e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + 4.60e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.94e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 1.31e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + 1.22e6T + 5.00e11T^{2} \) |
| 31 | \( 1 + 1.08e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 2.48e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + 2.64e4iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 4.08e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 6.95e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 8.77e6T + 6.22e13T^{2} \) |
| 59 | \( 1 - 3.74e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.40e7iT - 1.91e14T^{2} \) |
| 67 | \( 1 + 7.62e6T + 4.06e14T^{2} \) |
| 71 | \( 1 + 1.58e5T + 6.45e14T^{2} \) |
| 73 | \( 1 - 3.88e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 1.13e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 5.38e6iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 1.08e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 5.55e7iT - 7.83e15T^{2} \) |
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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
−10.02308049188289463221497678775, −8.998904050665343638921985188205, −8.180971120283319237151829994241, −7.51217752773638752914520037809, −5.56802361280068222302621518341, −5.18426189737029862274809028073, −4.07959401664533810986634901280, −2.42058128158586689602806215688, −1.20219532157759078359929802107, −0.25326076206467354298534150465,
1.61746315596018218818460770295, 2.62462644031150534959460644092, 3.80069724335421081996094944668, 5.03831845932564971966448846387, 6.30085310829130216097864485880, 7.23209383364461499874266739893, 7.979599637531170756754560389035, 9.263393822159665786570002744409, 10.46878089462267007501832900169, 11.00732746826877850457454869885
