| L(s) = 1 | + (−4.5 + 7.79i)3-s + (−11.2 − 19.4i)5-s + (96.4 + 86.5i)7-s + (−40.5 − 70.1i)9-s + (170. − 294. i)11-s − 728.·13-s + 202.·15-s + (−404. + 701. i)17-s + (513. + 888. i)19-s + (−1.10e3 + 362. i)21-s + (711. + 1.23e3i)23-s + (1.30e3 − 2.26e3i)25-s + 729·27-s + 5.21e3·29-s + (−3.51e3 + 6.09e3i)31-s + ⋯ |
| L(s) = 1 | + (−0.288 + 0.499i)3-s + (−0.201 − 0.348i)5-s + (0.744 + 0.667i)7-s + (−0.166 − 0.288i)9-s + (0.424 − 0.734i)11-s − 1.19·13-s + 0.232·15-s + (−0.339 + 0.588i)17-s + (0.326 + 0.564i)19-s + (−0.548 + 0.179i)21-s + (0.280 + 0.485i)23-s + (0.419 − 0.725i)25-s + 0.192·27-s + 1.15·29-s + (−0.657 + 1.13i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 - 0.267i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.963 - 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.6530976992\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6530976992\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (4.5 - 7.79i)T \) |
| 7 | \( 1 + (-96.4 - 86.5i)T \) |
| good | 5 | \( 1 + (11.2 + 19.4i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 11 | \( 1 + (-170. + 294. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + 728.T + 3.71e5T^{2} \) |
| 17 | \( 1 + (404. - 701. i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-513. - 888. i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-711. - 1.23e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 - 5.21e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (3.51e3 - 6.09e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (6.39e3 + 1.10e4i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 - 1.17e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 3.66e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-4.65e3 - 8.06e3i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (1.78e4 - 3.08e4i)T + (-2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (1.51e4 - 2.63e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (1.60e4 + 2.78e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-1.06e4 + 1.84e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + 6.11e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (2.06e4 - 3.57e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (1.75e4 + 3.03e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + 8.61e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (3.89e4 + 6.75e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + 1.61e5T + 8.58e9T^{2} \) |
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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
−11.12371789613013926100231121383, −10.33453607095052499775651833984, −9.140383908086825407899642182783, −8.542116378183797038172406321602, −7.43405440489018828499527026641, −6.08977406802697346458534352411, −5.16599322793603760425774586350, −4.30010729937356857222730498208, −2.91126666527624320595977150986, −1.39998275367882146143241893666,
0.17772440358612500032284825227, 1.53371413056489809555284023335, 2.80425232209686859272645099190, 4.42445899417816620148550033226, 5.16161245833028639427157226494, 6.84376097425132941238562797928, 7.17049658544612904984236278780, 8.204629951384130880292115367911, 9.465248455170271710804489528777, 10.39501857978055677288549894001
