| L(s) = 1 | + (−6.91 − 2.24i)2-s + (1.65 − 2.27i)3-s + (16.8 + 12.2i)4-s + (−34.0 − 44.3i)5-s + (−16.5 + 12.0i)6-s + 197. i·7-s + (47.7 + 65.7i)8-s + (72.6 + 223. i)9-s + (135. + 382. i)10-s + (51.7 − 159. i)11-s + (55.6 − 18.0i)12-s + (−936. + 304. i)13-s + (442. − 1.36e3i)14-s + (−157. + 4.23i)15-s + (−388. − 1.19e3i)16-s + (901. + 1.24e3i)17-s + ⋯ |
| L(s) = 1 | + (−1.22 − 0.397i)2-s + (0.106 − 0.145i)3-s + (0.526 + 0.382i)4-s + (−0.609 − 0.792i)5-s + (−0.187 + 0.136i)6-s + 1.52i·7-s + (0.263 + 0.363i)8-s + (0.298 + 0.920i)9-s + (0.429 + 1.21i)10-s + (0.129 − 0.397i)11-s + (0.111 − 0.0362i)12-s + (−1.53 + 0.499i)13-s + (0.603 − 1.85i)14-s + (−0.180 + 0.00486i)15-s + (−0.379 − 1.16i)16-s + (0.756 + 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00891 - 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.00891 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.274822 + 0.272385i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.274822 + 0.272385i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (34.0 + 44.3i)T \) |
| good | 2 | \( 1 + (6.91 + 2.24i)T + (25.8 + 18.8i)T^{2} \) |
| 3 | \( 1 + (-1.65 + 2.27i)T + (-75.0 - 231. i)T^{2} \) |
| 7 | \( 1 - 197. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (-51.7 + 159. i)T + (-1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (936. - 304. i)T + (3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-901. - 1.24e3i)T + (-4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-407. + 295. i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (3.41e3 + 1.10e3i)T + (5.20e6 + 3.78e6i)T^{2} \) |
| 29 | \( 1 + (278. + 202. i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (3.71e3 - 2.70e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (5.49e3 - 1.78e3i)T + (5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-352. - 1.08e3i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 - 1.12e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (4.05e3 - 5.57e3i)T + (-7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-2.29e4 + 3.15e4i)T + (-1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-301. - 927. i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-4.78e3 + 1.47e4i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + (-9.88e3 - 1.36e4i)T + (-4.17e8 + 1.28e9i)T^{2} \) |
| 71 | \( 1 + (2.18e4 + 1.58e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-3.54e4 - 1.15e4i)T + (1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-6.37e4 - 4.63e4i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (5.81e4 + 8.00e4i)T + (-1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 + (7.07e3 - 2.17e4i)T + (-4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (-7.05e4 + 9.71e4i)T + (-2.65e9 - 8.16e9i)T^{2} \) |
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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
−16.92645295289685745728868572065, −16.05346416138970624092679335354, −14.49713017243200798472990943526, −12.55390048745212541081656956941, −11.63105474041035436241841800489, −9.949676637844907784972118545840, −8.724636183484261686299306938356, −7.83544982844457311708259335182, −5.13971291784376556496043196684, −1.99582150616276027554367750539,
0.38255716092650478661843502207, 3.88856511315616939545994913523, 7.12979590659914374680360120467, 7.58707729544056366552203460463, 9.690906046966366192673604323920, 10.34612336164449841012717019068, 12.11708363612714316491773475899, 14.04753909176363285033100428208, 15.25307840906676249590462886991, 16.49315058571225163721998042267
