| L(s) = 1 | + (−1.54 − 3.68i)2-s + (12.3 − 2.18i)3-s + (−11.2 + 11.3i)4-s + (−26.3 + 22.0i)5-s + (−27.2 − 42.3i)6-s + (82.9 + 47.8i)7-s + (59.3 + 23.8i)8-s + (72.7 − 26.4i)9-s + (122. + 62.9i)10-s + (45.2 − 26.1i)11-s + (−114. + 165. i)12-s + (−18.4 + 104. i)13-s + (48.6 − 379. i)14-s + (−277. + 331. i)15-s + (−3.77 − 255. i)16-s + (172. + 62.7i)17-s + ⋯ |
| L(s) = 1 | + (−0.386 − 0.922i)2-s + (1.37 − 0.242i)3-s + (−0.701 + 0.712i)4-s + (−1.05 + 0.882i)5-s + (−0.755 − 1.17i)6-s + (1.69 + 0.977i)7-s + (0.928 + 0.372i)8-s + (0.898 − 0.326i)9-s + (1.22 + 0.629i)10-s + (0.373 − 0.215i)11-s + (−0.793 + 1.15i)12-s + (−0.108 + 0.617i)13-s + (0.247 − 1.93i)14-s + (−1.23 + 1.47i)15-s + (−0.0147 − 0.999i)16-s + (0.596 + 0.217i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.107i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.994 + 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.92535 - 0.103993i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.92535 - 0.103993i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.54 + 3.68i)T \) |
| 19 | \( 1 + (360. - 3.95i)T \) |
| good | 3 | \( 1 + (-12.3 + 2.18i)T + (76.1 - 27.7i)T^{2} \) |
| 5 | \( 1 + (26.3 - 22.0i)T + (108. - 615. i)T^{2} \) |
| 7 | \( 1 + (-82.9 - 47.8i)T + (1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-45.2 + 26.1i)T + (7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (18.4 - 104. i)T + (-2.68e4 - 9.76e3i)T^{2} \) |
| 17 | \( 1 + (-172. - 62.7i)T + (6.39e4 + 5.36e4i)T^{2} \) |
| 23 | \( 1 + (-430. + 512. i)T + (-4.85e4 - 2.75e5i)T^{2} \) |
| 29 | \( 1 + (-159. + 58.0i)T + (5.41e5 - 4.54e5i)T^{2} \) |
| 31 | \( 1 + (-1.32e3 - 764. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + 1.21e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + (426. + 2.41e3i)T + (-2.65e6 + 9.66e5i)T^{2} \) |
| 43 | \( 1 + (32.3 + 38.5i)T + (-5.93e5 + 3.36e6i)T^{2} \) |
| 47 | \( 1 + (2.66 + 7.31i)T + (-3.73e6 + 3.13e6i)T^{2} \) |
| 53 | \( 1 + (573. + 480. i)T + (1.37e6 + 7.77e6i)T^{2} \) |
| 59 | \( 1 + (-5.09 + 13.9i)T + (-9.28e6 - 7.78e6i)T^{2} \) |
| 61 | \( 1 + (-1.63e3 - 1.36e3i)T + (2.40e6 + 1.36e7i)T^{2} \) |
| 67 | \( 1 + (505. + 1.38e3i)T + (-1.54e7 + 1.29e7i)T^{2} \) |
| 71 | \( 1 + (4.48e3 + 5.34e3i)T + (-4.41e6 + 2.50e7i)T^{2} \) |
| 73 | \( 1 + (828. + 4.70e3i)T + (-2.66e7 + 9.71e6i)T^{2} \) |
| 79 | \( 1 + (4.67e3 - 824. i)T + (3.66e7 - 1.33e7i)T^{2} \) |
| 83 | \( 1 + (4.48e3 + 2.59e3i)T + (2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-1.04e3 + 5.91e3i)T + (-5.89e7 - 2.14e7i)T^{2} \) |
| 97 | \( 1 + (8.36e3 + 3.04e3i)T + (6.78e7 + 5.69e7i)T^{2} \) |
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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
−14.03093644199655237831121512772, −12.33647429731014005371975460376, −11.56143190624009255315530518046, −10.58752067648227450960448283743, −8.727421807069151903493555432468, −8.440143656819355275442854921022, −7.33250128998711430655474278451, −4.45697303381849232286418470486, −3.06634491906344283184028336706, −1.89127643086070036951161248819,
1.15048914738517990080788161891, 4.02544446130422331131902544183, 4.88059357790592803917418845037, 7.41990243653743039135776821941, 8.108304961023288644081862328566, 8.647355257321055093433475198333, 10.04298574343098914792159275252, 11.47837214563146962476767518350, 13.17800537954812827632123519600, 14.18513673542557509111541972430
