| L(s) = 1 | + (3.06 − 2.57i)2-s + (−15.1 − 12.6i)3-s + (2.77 − 15.7i)4-s + (77.8 − 28.3i)5-s − 78.8·6-s + (134. − 48.9i)7-s + (−32.0 − 55.4i)8-s + (25.3 + 143. i)9-s + (165. − 287. i)10-s + (−132. − 229. i)11-s + (−241. + 202. i)12-s + (−52.5 + 298. i)13-s + (286. − 496. i)14-s + (−1.53e3 − 559. i)15-s + (−240. − 87.5i)16-s + (57.5 + 326. i)17-s + ⋯ |
| L(s) = 1 | + (0.541 − 0.454i)2-s + (−0.969 − 0.813i)3-s + (0.0868 − 0.492i)4-s + (1.39 − 0.507i)5-s − 0.894·6-s + (1.03 − 0.377i)7-s + (−0.176 − 0.306i)8-s + (0.104 + 0.591i)9-s + (0.524 − 0.908i)10-s + (−0.330 − 0.572i)11-s + (−0.484 + 0.406i)12-s + (−0.0862 + 0.489i)13-s + (0.390 − 0.676i)14-s + (−1.76 − 0.641i)15-s + (−0.234 − 0.0855i)16-s + (0.0482 + 0.273i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.777 + 0.628i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.777 + 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.720579 - 2.03733i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.720579 - 2.03733i\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + (-4.50e3 - 7.00e3i)T \) |
| good | 3 | \( 1 + (15.1 + 12.6i)T + (42.1 + 239. i)T^{2} \) |
| 5 | \( 1 + (-77.8 + 28.3i)T + (2.39e3 - 2.00e3i)T^{2} \) |
| 7 | \( 1 + (-134. + 48.9i)T + (1.28e4 - 1.08e4i)T^{2} \) |
| 11 | \( 1 + (132. + 229. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (52.5 - 298. i)T + (-3.48e5 - 1.26e5i)T^{2} \) |
| 17 | \( 1 + (-57.5 - 326. i)T + (-1.33e6 + 4.85e5i)T^{2} \) |
| 19 | \( 1 + (1.80e3 + 1.51e3i)T + (4.29e5 + 2.43e6i)T^{2} \) |
| 23 | \( 1 + (506. - 876. i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-155. - 269. i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 - 8.92e3T + 2.86e7T^{2} \) |
| 41 | \( 1 + (-1.01e3 + 5.73e3i)T + (-1.08e8 - 3.96e7i)T^{2} \) |
| 43 | \( 1 + 1.88e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.18e4 + 2.04e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-1.18e4 - 4.30e3i)T + (3.20e8 + 2.68e8i)T^{2} \) |
| 59 | \( 1 + (-2.86e4 - 1.04e4i)T + (5.47e8 + 4.59e8i)T^{2} \) |
| 61 | \( 1 + (5.71e3 - 3.24e4i)T + (-7.93e8 - 2.88e8i)T^{2} \) |
| 67 | \( 1 + (-5.55e4 + 2.02e4i)T + (1.03e9 - 8.67e8i)T^{2} \) |
| 71 | \( 1 + (3.36e4 + 2.82e4i)T + (3.13e8 + 1.77e9i)T^{2} \) |
| 73 | \( 1 + 1.79e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (2.34e4 - 8.54e3i)T + (2.35e9 - 1.97e9i)T^{2} \) |
| 83 | \( 1 + (-6.67e3 - 3.78e4i)T + (-3.70e9 + 1.34e9i)T^{2} \) |
| 89 | \( 1 + (-1.02e5 - 3.73e4i)T + (4.27e9 + 3.58e9i)T^{2} \) |
| 97 | \( 1 + (-4.15e4 + 7.20e4i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−13.31566207113461921696771004809, −12.07436935094859284842212570889, −11.18492246087718772811431342861, −10.13905922864542922092472450304, −8.580707726833890722566714379092, −6.73497918505764740358901163339, −5.74323769762631428545601379518, −4.72607702035154157168863309296, −2.06576046612208240293209577767, −0.931500516708979915167308064538,
2.25156225739064749779289445241, 4.57079766336426438912169013815, 5.49841871571162700840262000840, 6.36785482747980645031870346947, 8.144713332401141874453492252250, 9.883193815268944779950345819901, 10.61425467684231433999851404425, 11.76045913538358036824893422680, 13.01598089158752145069736962307, 14.27419612757967710714997931533
