| L(s) = 1 | + (−0.923 − 0.923i)2-s − 30.2i·4-s + (31.9 − 31.9i)7-s + (−57.5 + 57.5i)8-s + 375. i·11-s + (−278. − 278. i)13-s − 59.0·14-s − 863.·16-s + (−812. − 812. i)17-s + 23.2i·19-s + (347. − 347. i)22-s + (−1.76e3 + 1.76e3i)23-s + 514. i·26-s + (−968. − 968. i)28-s + 4.94e3·29-s + ⋯ |
| L(s) = 1 | + (−0.163 − 0.163i)2-s − 0.946i·4-s + (0.246 − 0.246i)7-s + (−0.317 + 0.317i)8-s + 0.936i·11-s + (−0.456 − 0.456i)13-s − 0.0805·14-s − 0.842·16-s + (−0.681 − 0.681i)17-s + 0.0147i·19-s + (0.152 − 0.152i)22-s + (−0.694 + 0.694i)23-s + 0.149i·26-s + (−0.233 − 0.233i)28-s + 1.09·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.161 - 0.986i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.161 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4359268191\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4359268191\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + (0.923 + 0.923i)T + 32iT^{2} \) |
| 7 | \( 1 + (-31.9 + 31.9i)T - 1.68e4iT^{2} \) |
| 11 | \( 1 - 375. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (278. + 278. i)T + 3.71e5iT^{2} \) |
| 17 | \( 1 + (812. + 812. i)T + 1.41e6iT^{2} \) |
| 19 | \( 1 - 23.2iT - 2.47e6T^{2} \) |
| 23 | \( 1 + (1.76e3 - 1.76e3i)T - 6.43e6iT^{2} \) |
| 29 | \( 1 - 4.94e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 292.T + 2.86e7T^{2} \) |
| 37 | \( 1 + (8.12e3 - 8.12e3i)T - 6.93e7iT^{2} \) |
| 41 | \( 1 + 5.09e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (7.77e3 + 7.77e3i)T + 1.47e8iT^{2} \) |
| 47 | \( 1 + (-1.55e4 - 1.55e4i)T + 2.29e8iT^{2} \) |
| 53 | \( 1 + (2.62e4 - 2.62e4i)T - 4.18e8iT^{2} \) |
| 59 | \( 1 - 4.29e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.07e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (-3.97e3 + 3.97e3i)T - 1.35e9iT^{2} \) |
| 71 | \( 1 - 5.09e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (5.69e4 + 5.69e4i)T + 2.07e9iT^{2} \) |
| 79 | \( 1 - 6.90e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + (7.36e4 - 7.36e4i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 - 5.39e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (2.52e4 - 2.52e4i)T - 8.58e9iT^{2} \) |
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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
−11.55133680970795245134128845752, −10.50214972215490653294891235813, −9.863768153040763933840218607646, −8.908980932581677312714112437706, −7.58802349142172940501516721775, −6.58223977350174260602868711327, −5.32349542599738641998087753889, −4.42768827402270049910346659438, −2.56542825188603673976614106975, −1.32052082051804251374613563514,
0.13485805330280812187617984979, 2.16466148371731809168858941307, 3.47300143930494273735946366871, 4.63078420514757757942300895668, 6.13672121931209231805082413533, 7.11461104454570037667690492185, 8.362254603016582027566373720320, 8.746530395082359952528162900615, 10.11564642741824049410446659737, 11.28199268505379950556062184293
