| L(s) = 1 | + (4.25 + 5.85i)3-s + (9.99 + 5.01i)5-s + 4.97i·7-s + (−7.86 + 24.2i)9-s + (−7.23 − 22.2i)11-s + (8.80 + 2.85i)13-s + (13.1 + 79.9i)15-s + (−56.0 + 77.1i)17-s + (−40.4 − 29.3i)19-s + (−29.1 + 21.1i)21-s + (187. − 61.0i)23-s + (74.6 + 100. i)25-s + (10.6 − 3.46i)27-s + (−23.7 + 17.2i)29-s + (−121. − 88.5i)31-s + ⋯ |
| L(s) = 1 | + (0.819 + 1.12i)3-s + (0.893 + 0.448i)5-s + 0.268i·7-s + (−0.291 + 0.896i)9-s + (−0.198 − 0.609i)11-s + (0.187 + 0.0610i)13-s + (0.225 + 1.37i)15-s + (−0.800 + 1.10i)17-s + (−0.488 − 0.354i)19-s + (−0.302 + 0.220i)21-s + (1.70 − 0.553i)23-s + (0.596 + 0.802i)25-s + (0.0760 − 0.0247i)27-s + (−0.152 + 0.110i)29-s + (−0.705 − 0.512i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.237 - 0.971i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.237 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.74106 + 1.36643i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.74106 + 1.36643i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-9.99 - 5.01i)T \) |
| good | 3 | \( 1 + (-4.25 - 5.85i)T + (-8.34 + 25.6i)T^{2} \) |
| 7 | \( 1 - 4.97iT - 343T^{2} \) |
| 11 | \( 1 + (7.23 + 22.2i)T + (-1.07e3 + 782. i)T^{2} \) |
| 13 | \( 1 + (-8.80 - 2.85i)T + (1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (56.0 - 77.1i)T + (-1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (40.4 + 29.3i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 + (-187. + 61.0i)T + (9.84e3 - 7.15e3i)T^{2} \) |
| 29 | \( 1 + (23.7 - 17.2i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (121. + 88.5i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (50.4 + 16.3i)T + (4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-106. + 328. i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + 170. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (342. + 471. i)T + (-3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (45.4 + 62.5i)T + (-4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-25.4 + 78.4i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-183. - 565. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 + (-440. + 606. i)T + (-9.29e4 - 2.86e5i)T^{2} \) |
| 71 | \( 1 + (863. - 627. i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-781. + 253. i)T + (3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-387. + 281. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (495. - 682. i)T + (-1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + (-183. - 564. i)T + (-5.70e5 + 4.14e5i)T^{2} \) |
| 97 | \( 1 + (152. + 210. i)T + (-2.82e5 + 8.68e5i)T^{2} \) |
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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
−13.75459627392274686671860989843, −12.88609901363220559632811786934, −10.99158772978201697124467867584, −10.40969477670247134327820988532, −9.151181565686060696290702969639, −8.611902211384867622378913625260, −6.71141246670982767028004424593, −5.32049304163856650672419533861, −3.76933783758674828092036664489, −2.42776962890647278074424622126,
1.38538028244611713364542482112, 2.71294666062189323090558084653, 4.91493260386239036660127137533, 6.55642777229808689405976850822, 7.50946420421118417398398813739, 8.747872000146554905141809286445, 9.621202948383567282176790345001, 11.09206725873421753751845880430, 12.65289619929686672342328199268, 13.13790538521642406925141513909
