| L(s) = 1 | + (6.57 + 2.13i)2-s + (10.0 − 13.8i)3-s + (12.7 + 9.24i)4-s + (15.9 − 53.5i)5-s + (95.6 − 69.5i)6-s + 188. i·7-s + (−66.0 − 90.9i)8-s + (−15.4 − 47.4i)9-s + (218. − 318. i)10-s + (−215. + 661. i)11-s + (256. − 83.1i)12-s + (666. − 216. i)13-s + (−403. + 1.24e3i)14-s + (−581. − 759. i)15-s + (−395. − 1.21e3i)16-s + (−519. − 714. i)17-s + ⋯ |
| L(s) = 1 | + (1.16 + 0.377i)2-s + (0.645 − 0.888i)3-s + (0.397 + 0.288i)4-s + (0.284 − 0.958i)5-s + (1.08 − 0.788i)6-s + 1.45i·7-s + (−0.365 − 0.502i)8-s + (−0.0635 − 0.195i)9-s + (0.692 − 1.00i)10-s + (−0.535 + 1.64i)11-s + (0.513 − 0.166i)12-s + (1.09 − 0.355i)13-s + (−0.550 + 1.69i)14-s + (−0.667 − 0.871i)15-s + (−0.386 − 1.18i)16-s + (−0.435 − 0.599i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.82850 - 0.465111i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.82850 - 0.465111i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-15.9 + 53.5i)T \) |
| good | 2 | \( 1 + (-6.57 - 2.13i)T + (25.8 + 18.8i)T^{2} \) |
| 3 | \( 1 + (-10.0 + 13.8i)T + (-75.0 - 231. i)T^{2} \) |
| 7 | \( 1 - 188. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (215. - 661. i)T + (-1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (-666. + 216. i)T + (3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (519. + 714. i)T + (-4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (890. - 647. i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (1.67e3 + 545. i)T + (5.20e6 + 3.78e6i)T^{2} \) |
| 29 | \( 1 + (1.15e3 + 835. i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-4.81e3 + 3.50e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-3.65e3 + 1.18e3i)T + (5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (454. + 1.39e3i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 - 4.95e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (1.32e3 - 1.82e3i)T + (-7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (1.33e4 - 1.84e4i)T + (-1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (9.62e3 + 2.96e4i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-6.52e3 + 2.00e4i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + (1.89e4 + 2.60e4i)T + (-4.17e8 + 1.28e9i)T^{2} \) |
| 71 | \( 1 + (3.77e4 + 2.74e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-3.76e4 - 1.22e4i)T + (1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-6.53e4 - 4.75e4i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-3.33e4 - 4.59e4i)T + (-1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 + (-2.34e4 + 7.20e4i)T + (-4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (1.49e4 - 2.06e4i)T + (-2.65e9 - 8.16e9i)T^{2} \) |
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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
−15.90020725582858392873962573355, −15.10671950623025351355942786917, −13.72374258384776817644655953151, −12.80710010237001036349236342133, −12.25948092094187449869256635694, −9.423723956749157139699709519549, −8.064858902272226339006522674086, −6.16779204307580927187454720810, −4.77888194137444219014629029961, −2.21261822777104877092629927992,
3.21990795227462233957998862140, 4.08212765396731305331956376842, 6.26125238539904397590312392436, 8.525057595704096710530059616376, 10.44287701645271405282213463797, 11.16542896672735080751630278228, 13.49322995337749807338368936329, 13.82730811698484411592964531553, 14.95152098334170543392336948533, 16.20260241841033420303296813075
