| L(s) = 1 | + (19.2 − 33.2i)2-s + (97.9 + 169. i)3-s + (−482. − 835. i)4-s + (−39.7 − 68.8i)5-s + 7.52e3·6-s + (5.97e3 + 1.03e4i)7-s − 1.74e4·8-s + (−9.34e3 + 1.61e4i)9-s − 3.05e3·10-s − 1.07e4·11-s + (9.45e4 − 1.63e5i)12-s + (5.83e4 + 1.01e5i)13-s + 4.59e5·14-s + (7.78e3 − 1.34e4i)15-s + (−8.75e4 + 1.51e5i)16-s + (2.02e4 − 3.50e4i)17-s + ⋯ |
| L(s) = 1 | + (0.849 − 1.47i)2-s + (0.698 + 1.20i)3-s + (−0.942 − 1.63i)4-s + (−0.0284 − 0.0492i)5-s + 2.37·6-s + (0.940 + 1.62i)7-s − 1.50·8-s + (−0.474 + 0.822i)9-s − 0.0965·10-s − 0.222·11-s + (1.31 − 2.27i)12-s + (0.566 + 0.981i)13-s + 3.19·14-s + (0.0396 − 0.0687i)15-s + (−0.333 + 0.578i)16-s + (0.0588 − 0.101i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.268i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.963 + 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.84888 - 0.525861i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.84888 - 0.525861i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (1.19e6 + 1.13e7i)T \) |
| good | 2 | \( 1 + (-19.2 + 33.2i)T + (-256 - 443. i)T^{2} \) |
| 3 | \( 1 + (-97.9 - 169. i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (39.7 + 68.8i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 7 | \( 1 + (-5.97e3 - 1.03e4i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + 1.07e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + (-5.83e4 - 1.01e5i)T + (-5.30e9 + 9.18e9i)T^{2} \) |
| 17 | \( 1 + (-2.02e4 + 3.50e4i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-2.21e4 - 3.83e4i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 - 1.92e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 5.89e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.85e6T + 2.64e13T^{2} \) |
| 41 | \( 1 + (-9.41e6 - 1.63e7i)T + (-1.63e14 + 2.83e14i)T^{2} \) |
| 43 | \( 1 + 2.21e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.10e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + (-6.78e6 + 1.17e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-5.31e7 + 9.20e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-4.77e7 - 8.26e7i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-1.10e8 - 1.90e8i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 + (1.36e8 + 2.36e8i)T + (-2.29e16 + 3.97e16i)T^{2} \) |
| 73 | \( 1 - 1.53e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + (3.09e8 + 5.35e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (4.69e7 - 8.12e7i)T + (-9.34e16 - 1.61e17i)T^{2} \) |
| 89 | \( 1 + (-3.55e8 + 6.14e8i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 + 8.99e8T + 7.60e17T^{2} \) |
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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
−14.48504970625793302009158456076, −12.97697530885338843707267880855, −11.69109111757636328761487448743, −10.92333343112402284595249222679, −9.465763229513013768909485411466, −8.695772698756875682195188935913, −5.38415314158328079969725339787, −4.39234831188718470917167191576, −3.03463820932227953659788611861, −1.86857192735595412412734297441,
1.16944200844835795880950334915, 3.56119868393444455407250285925, 5.17261431524687312717354831119, 6.93977007159085758573513327636, 7.53923328398545379887238967312, 8.411698337425932294721643901071, 10.91738488297433573324071525714, 13.02574833480471429968852612977, 13.34393172207885346558974580449, 14.36301335727685473124586622000
