| L(s) = 1 | + (3.06 − 2.57i)2-s + (21.8 + 18.3i)3-s + (2.77 − 15.7i)4-s + (−84.9 + 30.9i)5-s + 114.·6-s + (−181. + 65.9i)7-s + (−32.0 − 55.4i)8-s + (98.9 + 561. i)9-s + (−180. + 313. i)10-s + (234. + 406. i)11-s + (349. − 293. i)12-s + (69.1 − 392. i)13-s + (−385. + 667. i)14-s + (−2.42e3 − 881. i)15-s + (−240. − 87.5i)16-s + (−113. − 645. i)17-s + ⋯ |
| L(s) = 1 | + (0.541 − 0.454i)2-s + (1.40 + 1.17i)3-s + (0.0868 − 0.492i)4-s + (−1.51 + 0.552i)5-s + 1.29·6-s + (−1.39 + 0.508i)7-s + (−0.176 − 0.306i)8-s + (0.407 + 2.31i)9-s + (−0.571 + 0.989i)10-s + (0.584 + 1.01i)11-s + (0.700 − 0.587i)12-s + (0.113 − 0.643i)13-s + (−0.525 + 0.910i)14-s + (−2.77 − 1.01i)15-s + (−0.234 − 0.0855i)16-s + (−0.0955 − 0.541i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.328 - 0.944i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.27199 + 1.78829i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27199 + 1.78829i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.06 + 2.57i)T \) |
| 37 | \( 1 + (8.11e3 - 1.86e3i)T \) |
| good | 3 | \( 1 + (-21.8 - 18.3i)T + (42.1 + 239. i)T^{2} \) |
| 5 | \( 1 + (84.9 - 30.9i)T + (2.39e3 - 2.00e3i)T^{2} \) |
| 7 | \( 1 + (181. - 65.9i)T + (1.28e4 - 1.08e4i)T^{2} \) |
| 11 | \( 1 + (-234. - 406. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-69.1 + 392. i)T + (-3.48e5 - 1.26e5i)T^{2} \) |
| 17 | \( 1 + (113. + 645. i)T + (-1.33e6 + 4.85e5i)T^{2} \) |
| 19 | \( 1 + (-1.50e3 - 1.26e3i)T + (4.29e5 + 2.43e6i)T^{2} \) |
| 23 | \( 1 + (-1.02e3 + 1.77e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-2.76e3 - 4.79e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 - 5.05e3T + 2.86e7T^{2} \) |
| 41 | \( 1 + (2.91e3 - 1.65e4i)T + (-1.08e8 - 3.96e7i)T^{2} \) |
| 43 | \( 1 + 3.01e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-2.80e3 + 4.86e3i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (2.10e4 + 7.64e3i)T + (3.20e8 + 2.68e8i)T^{2} \) |
| 59 | \( 1 + (-2.15e4 - 7.86e3i)T + (5.47e8 + 4.59e8i)T^{2} \) |
| 61 | \( 1 + (-1.72e3 + 9.80e3i)T + (-7.93e8 - 2.88e8i)T^{2} \) |
| 67 | \( 1 + (-3.35e3 + 1.22e3i)T + (1.03e9 - 8.67e8i)T^{2} \) |
| 71 | \( 1 + (-5.58e4 - 4.68e4i)T + (3.13e8 + 1.77e9i)T^{2} \) |
| 73 | \( 1 + 4.35e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (1.88e4 - 6.87e3i)T + (2.35e9 - 1.97e9i)T^{2} \) |
| 83 | \( 1 + (1.78e4 + 1.01e5i)T + (-3.70e9 + 1.34e9i)T^{2} \) |
| 89 | \( 1 + (-2.94e4 - 1.07e4i)T + (4.27e9 + 3.58e9i)T^{2} \) |
| 97 | \( 1 + (1.25e4 - 2.18e4i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−14.23030317924683425454744026516, −12.82514632581501015869336025023, −11.80058473369193821130100032255, −10.34109336820744190402697371835, −9.617811814656202373652625109265, −8.373815599215111444326821891252, −6.98591940318661131864837502151, −4.66867687371064484222384488727, −3.45343594072923500774240447261, −2.95984724576865327104332846515,
0.69816208490494782817495397593, 3.19674452735504801201939077883, 3.89331530471554407107012083855, 6.53186562977314428217983539609, 7.33726541116505734610537087674, 8.373987480717069899624949930787, 9.222245065248785271139356749739, 11.65879334444233571635826819700, 12.47681453911398392798855231800, 13.47295737165562435165445036315
