| L(s) = 1 | + (3.33 − 6.55i)2-s + (−0.874 + 5.51i)3-s + (−22.4 − 30.8i)4-s + (2.16 − 24.9i)5-s + (33.2 + 24.1i)6-s + (37.7 + 37.7i)7-s + (−160. + 25.4i)8-s + (47.3 + 15.3i)9-s + (−156. − 97.3i)10-s + (17.9 + 55.3i)11-s + (189. − 96.7i)12-s + (58.0 + 113. i)13-s + (373. − 121. i)14-s + (135. + 33.6i)15-s + (−181. + 558. i)16-s + (−77.3 − 488. i)17-s + ⋯ |
| L(s) = 1 | + (0.834 − 1.63i)2-s + (−0.0971 + 0.613i)3-s + (−1.40 − 1.92i)4-s + (0.0864 − 0.996i)5-s + (0.923 + 0.671i)6-s + (0.770 + 0.770i)7-s + (−2.51 + 0.397i)8-s + (0.584 + 0.189i)9-s + (−1.56 − 0.973i)10-s + (0.148 + 0.457i)11-s + (1.31 − 0.671i)12-s + (0.343 + 0.674i)13-s + (1.90 − 0.619i)14-s + (0.602 + 0.149i)15-s + (−0.708 + 2.18i)16-s + (−0.267 − 1.69i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.411 + 0.911i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.411 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.01427 - 1.57137i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01427 - 1.57137i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-2.16 + 24.9i)T \) |
| good | 2 | \( 1 + (-3.33 + 6.55i)T + (-9.40 - 12.9i)T^{2} \) |
| 3 | \( 1 + (0.874 - 5.51i)T + (-77.0 - 25.0i)T^{2} \) |
| 7 | \( 1 + (-37.7 - 37.7i)T + 2.40e3iT^{2} \) |
| 11 | \( 1 + (-17.9 - 55.3i)T + (-1.18e4 + 8.60e3i)T^{2} \) |
| 13 | \( 1 + (-58.0 - 113. i)T + (-1.67e4 + 2.31e4i)T^{2} \) |
| 17 | \( 1 + (77.3 + 488. i)T + (-7.94e4 + 2.58e4i)T^{2} \) |
| 19 | \( 1 + (315. - 433. i)T + (-4.02e4 - 1.23e5i)T^{2} \) |
| 23 | \( 1 + (-20.2 - 10.3i)T + (1.64e5 + 2.26e5i)T^{2} \) |
| 29 | \( 1 + (-126. - 173. i)T + (-2.18e5 + 6.72e5i)T^{2} \) |
| 31 | \( 1 + (-76.0 - 55.2i)T + (2.85e5 + 8.78e5i)T^{2} \) |
| 37 | \( 1 + (675. - 344. i)T + (1.10e6 - 1.51e6i)T^{2} \) |
| 41 | \( 1 + (549. - 1.69e3i)T + (-2.28e6 - 1.66e6i)T^{2} \) |
| 43 | \( 1 + (-1.86e3 + 1.86e3i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 + (1.11e3 + 176. i)T + (4.64e6 + 1.50e6i)T^{2} \) |
| 53 | \( 1 + (-441. + 2.78e3i)T + (-7.50e6 - 2.43e6i)T^{2} \) |
| 59 | \( 1 + (-2.17e3 - 708. i)T + (9.80e6 + 7.12e6i)T^{2} \) |
| 61 | \( 1 + (-1.44e3 - 4.44e3i)T + (-1.12e7 + 8.13e6i)T^{2} \) |
| 67 | \( 1 + (676. + 4.27e3i)T + (-1.91e7 + 6.22e6i)T^{2} \) |
| 71 | \( 1 + (1.89e3 - 1.37e3i)T + (7.85e6 - 2.41e7i)T^{2} \) |
| 73 | \( 1 + (3.60e3 + 1.83e3i)T + (1.66e7 + 2.29e7i)T^{2} \) |
| 79 | \( 1 + (3.58e3 + 4.93e3i)T + (-1.20e7 + 3.70e7i)T^{2} \) |
| 83 | \( 1 + (-8.36e3 + 1.32e3i)T + (4.51e7 - 1.46e7i)T^{2} \) |
| 89 | \( 1 + (1.89e3 - 616. i)T + (5.07e7 - 3.68e7i)T^{2} \) |
| 97 | \( 1 + (1.22e4 + 1.93e3i)T + (8.41e7 + 2.73e7i)T^{2} \) |
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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
−16.23957709710510811953571888962, −14.82765989251189952397955758393, −13.54886722190561562833802423931, −12.32598671359863147684319219855, −11.49644841652294928193888124167, −10.03528689495039181491423939549, −8.939217428443407795162789670366, −5.20524784165806593590471331226, −4.28676392512935035482273462207, −1.79095612805226130313134898946,
4.07619979424830735258693156620, 6.13451637451892862143245482311, 7.12462643366041962860954093661, 8.242319721037895452945625623155, 10.82258395855767790220245571945, 12.80414357411411655574472647405, 13.71153757056931692041456870490, 14.78056447947213948444436334857, 15.61706101390297796421737246894, 17.35424179230977791514074171287
