| L(s) = 1 | + (−3.55 + 1.83i)2-s + (−12.3 + 2.18i)3-s + (9.27 − 13.0i)4-s + (−26.3 + 22.0i)5-s + (40.0 − 30.5i)6-s + (−82.9 − 47.8i)7-s + (−9.05 + 63.3i)8-s + (72.7 − 26.4i)9-s + (53.0 − 126. i)10-s + (−45.2 + 26.1i)11-s + (−86.4 + 181. i)12-s + (−18.4 + 104. i)13-s + (382. + 18.1i)14-s + (277. − 331. i)15-s + (−84.0 − 241. i)16-s + (172. + 62.7i)17-s + ⋯ |
| L(s) = 1 | + (−0.888 + 0.458i)2-s + (−1.37 + 0.242i)3-s + (0.579 − 0.814i)4-s + (−1.05 + 0.882i)5-s + (1.11 − 0.847i)6-s + (−1.69 − 0.977i)7-s + (−0.141 + 0.989i)8-s + (0.898 − 0.326i)9-s + (0.530 − 1.26i)10-s + (−0.373 + 0.215i)11-s + (−0.600 + 1.26i)12-s + (−0.108 + 0.617i)13-s + (1.95 + 0.0924i)14-s + (1.23 − 1.47i)15-s + (−0.328 − 0.944i)16-s + (0.596 + 0.217i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0518i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.998 - 0.0518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.175576 + 0.00455514i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.175576 + 0.00455514i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.55 - 1.83i)T \) |
| 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
−13.94738997918452138596222091654, −12.23046239980230622258079363965, −11.27511995153695917702907674958, −10.40123927092804558671574020623, −9.651488892867981985272192138787, −7.47098487024453396387460458794, −6.88615205896472230210345376370, −5.70841049097749631324944060301, −3.64482932635093475012071263179, −0.29608684868813346064590567658,
0.56426501414537167307117469320, 3.27693559372293315865855648360, 5.41470384689467203686942429528, 6.69081337734493014735692180914, 8.098103226920700009107490888247, 9.339686471964003054762114241299, 10.48228389847575741103699302648, 11.79894415621608374320571566886, 12.33202327569977950035865661214, 12.85933509329382468525409080026
