| L(s) = 1 | + (−0.447 + 1.37i)2-s + (−17.2 − 12.5i)3-s + (24.1 + 17.5i)4-s + (14.3 + 54.0i)5-s + (24.9 − 18.1i)6-s + 250.·7-s + (−72.5 + 52.7i)8-s + (65.1 + 200. i)9-s + (−80.8 − 4.37i)10-s + (−170. + 523. i)11-s + (−196. − 605. i)12-s + (−92.7 − 285. i)13-s + (−112. + 345. i)14-s + (428. − 1.11e3i)15-s + (255. + 786. i)16-s + (637. − 463. i)17-s + ⋯ |
| L(s) = 1 | + (−0.0791 + 0.243i)2-s + (−1.10 − 0.803i)3-s + (0.755 + 0.549i)4-s + (0.257 + 0.966i)5-s + (0.283 − 0.205i)6-s + 1.93·7-s + (−0.400 + 0.291i)8-s + (0.268 + 0.825i)9-s + (−0.255 − 0.0138i)10-s + (−0.423 + 1.30i)11-s + (−0.394 − 1.21i)12-s + (−0.152 − 0.468i)13-s + (−0.152 + 0.470i)14-s + (0.491 − 1.27i)15-s + (0.249 + 0.768i)16-s + (0.535 − 0.388i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.25258 + 0.508742i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25258 + 0.508742i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-14.3 - 54.0i)T \) |
| good | 2 | \( 1 + (0.447 - 1.37i)T + (-25.8 - 18.8i)T^{2} \) |
| 3 | \( 1 + (17.2 + 12.5i)T + (75.0 + 231. i)T^{2} \) |
| 7 | \( 1 - 250.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (170. - 523. i)T + (-1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (92.7 + 285. i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-637. + 463. i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-432. + 314. i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (-252. + 777. i)T + (-5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-810. - 589. i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-1.56e3 + 1.13e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (4.13e3 + 1.27e4i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (3.66e3 + 1.12e4i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 + 7.12e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-6.17e3 - 4.48e3i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (1.62e4 + 1.18e4i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (1.15e3 + 3.56e3i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-3.13e3 + 9.63e3i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + (-3.08e4 + 2.23e4i)T + (4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (-2.10e4 - 1.53e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-5.96e3 + 1.83e4i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (4.74e4 + 3.44e4i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (6.77e4 - 4.92e4i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (5.73e3 - 1.76e4i)T + (-4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (-6.62e3 - 4.81e3i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−17.26927592256948780019167222041, −15.42389589614970930386198174742, −14.37756783809630697648569004132, −12.45447581672987063768179014476, −11.53590191738343198993663118163, −10.65630876521358654899472989260, −7.76114363027864992110362545237, −7.02814051001044802145683586207, −5.36654809447624634175871023802, −2.01138158014840650943553252660,
1.26125642613287895015667030781, 4.87580889375628582802523843856, 5.76158406167369993110368408898, 8.282785129088882683848353688402, 10.14749890589703913478395785732, 11.26371500950213309028569016333, 11.83526853476199739243744288281, 14.02668585624773181157585693404, 15.43015390034273328913167366487, 16.52422631303919314385171894276
