| L(s) = 1 | + (−0.722 + 2.22i)2-s + (−2.42 − 1.76i)3-s + (2.04 + 1.48i)4-s + (−1.87 − 11.0i)5-s + (5.67 − 4.12i)6-s − 32.9·7-s + (−19.9 + 14.4i)8-s + (2.78 + 8.55i)9-s + (25.8 + 3.78i)10-s + (15.3 − 47.0i)11-s + (−2.34 − 7.20i)12-s + (−16.0 − 49.4i)13-s + (23.8 − 73.2i)14-s + (−14.8 + 30.0i)15-s + (−11.5 − 35.5i)16-s + (47.8 − 34.7i)17-s + ⋯ |
| L(s) = 1 | + (−0.255 + 0.786i)2-s + (−0.467 − 0.339i)3-s + (0.255 + 0.185i)4-s + (−0.168 − 0.985i)5-s + (0.386 − 0.280i)6-s − 1.77·7-s + (−0.880 + 0.639i)8-s + (0.103 + 0.317i)9-s + (0.818 + 0.119i)10-s + (0.419 − 1.29i)11-s + (−0.0563 − 0.173i)12-s + (−0.342 − 1.05i)13-s + (0.454 − 1.39i)14-s + (−0.255 + 0.517i)15-s + (−0.180 − 0.555i)16-s + (0.682 − 0.495i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.345 + 0.938i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.345 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.216466 - 0.310527i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.216466 - 0.310527i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.42 + 1.76i)T \) |
| 5 | \( 1 + (1.87 + 11.0i)T \) |
| good | 2 | \( 1 + (0.722 - 2.22i)T + (-6.47 - 4.70i)T^{2} \) |
| 7 | \( 1 + 32.9T + 343T^{2} \) |
| 11 | \( 1 + (-15.3 + 47.0i)T + (-1.07e3 - 782. i)T^{2} \) |
| 13 | \( 1 + (16.0 + 49.4i)T + (-1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (-47.8 + 34.7i)T + (1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (118. - 85.8i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + (15.0 - 46.2i)T + (-9.84e3 - 7.15e3i)T^{2} \) |
| 29 | \( 1 + (-26.4 - 19.2i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-66.1 + 48.0i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-86.0 - 264. i)T + (-4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (91.7 + 282. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 + 64.8T + 7.95e4T^{2} \) |
| 47 | \( 1 + (124. + 90.6i)T + (3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (127. + 92.4i)T + (4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-126. - 389. i)T + (-1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (50.8 - 156. i)T + (-1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 + (-120. + 87.6i)T + (9.29e4 - 2.86e5i)T^{2} \) |
| 71 | \( 1 + (356. + 259. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-291. + 897. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (317. + 230. i)T + (1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-922. + 670. i)T + (1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + (-412. + 1.26e3i)T + (-5.70e5 - 4.14e5i)T^{2} \) |
| 97 | \( 1 + (-87.6 - 63.6i)T + (2.82e5 + 8.68e5i)T^{2} \) |
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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
−13.50473140121334595725668661759, −12.57553609659123646236082672703, −11.85538381318954454991011957782, −10.19034473528400488869170700110, −8.843219152174558997182807512414, −7.82749865962159180478412189591, −6.39371567324165523308163205321, −5.69348675129531961714411137356, −3.31776476362022008362124736033, −0.25632144848117172234860231318,
2.48704916716072680019305927197, 3.96147083247907059910350192467, 6.45659734123936710923941021586, 6.78014190038809411789093009110, 9.431422285286077878538894456845, 9.984828356738783742839204369226, 10.92068059214132269610217478344, 12.06724242757481158039145064610, 12.81564643584585775645081561544, 14.62137929903596215719318064316
