| L(s) = 1 | + (−0.642 − 0.766i)2-s + (1.43 + 1.20i)3-s + (−0.173 + 0.984i)4-s + (0.273 + 0.751i)5-s − 1.87i·6-s + (−0.138 + 0.0503i)7-s + (0.866 − 0.500i)8-s + (0.0923 + 0.524i)9-s + (0.400 − 0.692i)10-s + (−2.40 − 4.16i)11-s + (−1.43 + 1.20i)12-s + (1.91 + 0.338i)13-s + (0.127 + 0.0736i)14-s + (−0.514 + 1.41i)15-s + (−0.939 − 0.342i)16-s + (−3.43 + 0.606i)17-s + ⋯ |
| L(s) = 1 | + (−0.454 − 0.541i)2-s + (0.831 + 0.697i)3-s + (−0.0868 + 0.492i)4-s + (0.122 + 0.336i)5-s − 0.767i·6-s + (−0.0523 + 0.0190i)7-s + (0.306 − 0.176i)8-s + (0.0307 + 0.174i)9-s + (0.126 − 0.219i)10-s + (−0.725 − 1.25i)11-s + (−0.415 + 0.348i)12-s + (0.532 + 0.0938i)13-s + (0.0341 + 0.0196i)14-s + (−0.132 + 0.364i)15-s + (−0.234 − 0.0855i)16-s + (−0.833 + 0.147i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0176i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0176i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.919417 + 0.00809793i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.919417 + 0.00809793i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.642 + 0.766i)T \) |
| 37 | \( 1 + (-6.08 - 0.0772i)T \) |
| good | 3 | \( 1 + (-1.43 - 1.20i)T + (0.520 + 2.95i)T^{2} \) |
| 5 | \( 1 + (-0.273 - 0.751i)T + (-3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (0.138 - 0.0503i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (2.40 + 4.16i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.91 - 0.338i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (3.43 - 0.606i)T + (15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (4.33 - 5.16i)T + (-3.29 - 18.7i)T^{2} \) |
| 23 | \( 1 + (3.61 + 2.08i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.63 + 1.51i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 8.13iT - 31T^{2} \) |
| 41 | \( 1 + (-0.676 + 3.83i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + 8.10iT - 43T^{2} \) |
| 47 | \( 1 + (4.16 - 7.22i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-10.2 - 3.72i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (2.96 - 8.14i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (0.346 + 0.0610i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (2.25 - 0.820i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-5.34 - 4.48i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 - 1.13T + 73T^{2} \) |
| 79 | \( 1 + (-0.646 - 1.77i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (1.71 + 9.75i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (3.45 - 9.50i)T + (-68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-5.08 - 2.93i)T + (48.5 + 84.0i)T^{2} \) |
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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
−14.48676697163891012148697218928, −13.64770922964892457903698426834, −12.35276669500605563538676685014, −10.84462595652273885885302828756, −10.24656714723698800709143509419, −8.817185434793136564549391203793, −8.252563770138340291586051909302, −6.25434991670005036183902371270, −4.06126131264194166663343268673, −2.76378722663148320127200442087,
2.21904528216493128998787109743, 4.79178088602117717488276735476, 6.61823967574517458605710264106, 7.71294215514688841880146913024, 8.642476338286310745408090325821, 9.748204993650154907114700562170, 11.15261969697524827462867311359, 12.99272645919101583618772894828, 13.30577458533168828841334291439, 14.73746883422659812102545661012
