| 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
−14.62137929903596215719318064316, −12.81564643584585775645081561544, −12.06724242757481158039145064610, −10.92068059214132269610217478344, −9.984828356738783742839204369226, −9.431422285286077878538894456845, −6.78014190038809411789093009110, −6.45659734123936710923941021586, −3.96147083247907059910350192467, −2.48704916716072680019305927197,
0.25632144848117172234860231318, 3.31776476362022008362124736033, 5.69348675129531961714411137356, 6.39371567324165523308163205321, 7.82749865962159180478412189591, 8.843219152174558997182807512414, 10.19034473528400488869170700110, 11.85538381318954454991011957782, 12.57553609659123646236082672703, 13.50473140121334595725668661759
