| L(s) = 1 | + (−24.8 − 10.6i)3-s + (−81.8 − 141. i)5-s + (−30.8 − 17.8i)7-s + (502. + 528. i)9-s + (2.21e3 + 1.27e3i)11-s + (1.00e3 + 1.74e3i)13-s + (521. + 4.38e3i)15-s − 3.23e3·17-s − 8.56e3i·19-s + (576. + 770. i)21-s + (6.95e3 − 4.01e3i)23-s + (−5.57e3 + 9.65e3i)25-s + (−6.83e3 − 1.84e4i)27-s + (−1.83e4 + 3.16e4i)29-s + (2.14e4 − 1.24e4i)31-s + ⋯ |
| L(s) = 1 | + (−0.918 − 0.394i)3-s + (−0.654 − 1.13i)5-s + (−0.0900 − 0.0519i)7-s + (0.689 + 0.724i)9-s + (1.66 + 0.959i)11-s + (0.458 + 0.794i)13-s + (0.154 + 1.29i)15-s − 0.657·17-s − 1.24i·19-s + (0.0622 + 0.0832i)21-s + (0.571 − 0.329i)23-s + (−0.356 + 0.618i)25-s + (−0.347 − 0.937i)27-s + (−0.750 + 1.29i)29-s + (0.721 − 0.416i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0619 + 0.998i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.0619 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.228128222\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.228128222\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (24.8 + 10.6i)T \) |
| good | 5 | \( 1 + (81.8 + 141. i)T + (-7.81e3 + 1.35e4i)T^{2} \) |
| 7 | \( 1 + (30.8 + 17.8i)T + (5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-2.21e3 - 1.27e3i)T + (8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (-1.00e3 - 1.74e3i)T + (-2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 + 3.23e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + 8.56e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + (-6.95e3 + 4.01e3i)T + (7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (1.83e4 - 3.16e4i)T + (-2.97e8 - 5.15e8i)T^{2} \) |
| 31 | \( 1 + (-2.14e4 + 1.24e4i)T + (4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 - 8.23e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + (3.66e4 + 6.34e4i)T + (-2.37e9 + 4.11e9i)T^{2} \) |
| 43 | \( 1 + (-5.04e4 - 2.91e4i)T + (3.16e9 + 5.47e9i)T^{2} \) |
| 47 | \( 1 + (2.47e4 + 1.43e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 + 1.36e5T + 2.21e10T^{2} \) |
| 59 | \( 1 + (-2.53e5 + 1.46e5i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (4.83e4 - 8.36e4i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (8.25e4 - 4.76e4i)T + (4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 + 3.88e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 4.90e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + (-5.58e5 - 3.22e5i)T + (1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (2.22e5 + 1.28e5i)T + (1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 - 3.65e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + (-7.72e5 + 1.33e6i)T + (-4.16e11 - 7.21e11i)T^{2} \) |
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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
−11.73330013932841116114852003543, −11.14736671312968033833490812960, −9.482036222705599774928071893370, −8.709820153894066305894900952926, −7.19450733395086437094952633544, −6.44727738952927528519153467608, −4.81605167590650468748655144650, −4.19380929644309359623717722406, −1.66195606239573351864492902758, −0.57853486141165719507382780598,
0.978332703872130611479031640073, 3.29484352125584111541841562423, 4.14323299779067127725445671784, 5.96382616045973740869925752334, 6.53513041830651270925543277830, 7.86832372666223065409623458832, 9.297181228008091107491624189088, 10.43734431578928898740950098193, 11.35323242750009552637997595150, 11.71318268331444558954430193946
