| L(s) = 1 | + (6.91 + 9.51i)2-s + (−22.9 + 70.5i)4-s + (−174. − 127. i)5-s + (472. + 153. i)7-s + (−114. + 37.0i)8-s − 2.54e3i·10-s + (−360. − 1.28e3i)11-s + (−760. − 1.04e3i)13-s + (1.80e3 + 5.55e3i)14-s + (2.70e3 + 1.96e3i)16-s + (1.36e3 − 1.87e3i)17-s + (1.11e4 − 3.60e3i)19-s + (1.29e4 − 9.43e3i)20-s + (9.69e3 − 1.22e4i)22-s + 1.11e4·23-s + ⋯ |
| L(s) = 1 | + (0.863 + 1.18i)2-s + (−0.358 + 1.10i)4-s + (−1.39 − 1.01i)5-s + (1.37 + 0.447i)7-s + (−0.222 + 0.0724i)8-s − 2.54i·10-s + (−0.270 − 0.962i)11-s + (−0.346 − 0.476i)13-s + (0.657 + 2.02i)14-s + (0.659 + 0.479i)16-s + (0.277 − 0.381i)17-s + (1.61 − 0.526i)19-s + (1.62 − 1.17i)20-s + (0.910 − 1.15i)22-s + 0.914·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.336i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.941 - 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.71060 + 0.470079i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.71060 + 0.470079i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (360. + 1.28e3i)T \) |
| good | 2 | \( 1 + (-6.91 - 9.51i)T + (-19.7 + 60.8i)T^{2} \) |
| 5 | \( 1 + (174. + 127. i)T + (4.82e3 + 1.48e4i)T^{2} \) |
| 7 | \( 1 + (-472. - 153. i)T + (9.51e4 + 6.91e4i)T^{2} \) |
| 13 | \( 1 + (760. + 1.04e3i)T + (-1.49e6 + 4.59e6i)T^{2} \) |
| 17 | \( 1 + (-1.36e3 + 1.87e3i)T + (-7.45e6 - 2.29e7i)T^{2} \) |
| 19 | \( 1 + (-1.11e4 + 3.60e3i)T + (3.80e7 - 2.76e7i)T^{2} \) |
| 23 | \( 1 - 1.11e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + (3.16e3 + 1.02e3i)T + (4.81e8 + 3.49e8i)T^{2} \) |
| 31 | \( 1 + (1.45e4 - 1.05e4i)T + (2.74e8 - 8.44e8i)T^{2} \) |
| 37 | \( 1 + (-2.17e4 + 6.68e4i)T + (-2.07e9 - 1.50e9i)T^{2} \) |
| 41 | \( 1 + (4.14e4 - 1.34e4i)T + (3.84e9 - 2.79e9i)T^{2} \) |
| 43 | \( 1 + 7.93e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + (1.48e4 + 4.56e4i)T + (-8.72e9 + 6.33e9i)T^{2} \) |
| 53 | \( 1 + (7.63e3 - 5.54e3i)T + (6.84e9 - 2.10e10i)T^{2} \) |
| 59 | \( 1 + (-8.71e4 + 2.68e5i)T + (-3.41e10 - 2.47e10i)T^{2} \) |
| 61 | \( 1 + (-1.01e5 + 1.39e5i)T + (-1.59e10 - 4.89e10i)T^{2} \) |
| 67 | \( 1 + 4.60e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + (-2.05e5 - 1.49e5i)T + (3.95e10 + 1.21e11i)T^{2} \) |
| 73 | \( 1 + (-3.17e5 - 1.03e5i)T + (1.22e11 + 8.89e10i)T^{2} \) |
| 79 | \( 1 + (-6.21e4 - 8.55e4i)T + (-7.51e10 + 2.31e11i)T^{2} \) |
| 83 | \( 1 + (-2.02e5 + 2.79e5i)T + (-1.01e11 - 3.10e11i)T^{2} \) |
| 89 | \( 1 + 2.04e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + (-1.16e5 + 8.47e4i)T + (2.57e11 - 7.92e11i)T^{2} \) |
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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
−12.93915223990604300577447928623, −11.90397854633310071559398446749, −11.10796164921394472380968535545, −8.870347618439878977701205764227, −7.973340981369393668862484858398, −7.36165940270731446674775161661, −5.27389027281841945581188811313, −5.03727178914203276410553805924, −3.56625214684037724968923016031, −0.837597630679672458491535140402,
1.40721415162571390961983457820, 2.94772432978833271252732557330, 4.09799534841480769410399068672, 4.96634190205550775759481168727, 7.28664439677555651763709336776, 7.87721523863598260935511745947, 10.01147534564656616251273120687, 11.00816877294611740910317381303, 11.59711730234455813170556681800, 12.26292832206957811286520826972
