L(s) = 1 | + (−3.61 + 2.08i)2-s + (−3.72 + 6.45i)3-s + (4.70 − 8.15i)4-s + (−2.02 − 1.16i)5-s − 31.1i·6-s + (3.33 − 5.77i)7-s + 5.90i·8-s + (−14.3 − 24.7i)9-s + 9.76·10-s − 62.1·11-s + (35.1 + 60.8i)12-s + (43.8 + 25.3i)13-s + 27.8i·14-s + (15.1 − 8.72i)15-s + (25.3 + 43.8i)16-s + (−37.8 + 21.8i)17-s + ⋯ |
L(s) = 1 | + (−1.27 + 0.737i)2-s + (−0.717 + 1.24i)3-s + (0.588 − 1.01i)4-s + (−0.181 − 0.104i)5-s − 2.11i·6-s + (0.179 − 0.311i)7-s + 0.260i·8-s + (−0.530 − 0.918i)9-s + 0.308·10-s − 1.70·11-s + (0.844 + 1.46i)12-s + (0.935 + 0.540i)13-s + 0.531i·14-s + (0.260 − 0.150i)15-s + (0.395 + 0.685i)16-s + (−0.540 + 0.311i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.488 + 0.872i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0803545 - 0.137095i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0803545 - 0.137095i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 + (-216. - 62.4i)T \) |
good | 2 | \( 1 + (3.61 - 2.08i)T + (4 - 6.92i)T^{2} \) |
| 3 | \( 1 + (3.72 - 6.45i)T + (-13.5 - 23.3i)T^{2} \) |
| 5 | \( 1 + (2.02 + 1.16i)T + (62.5 + 108. i)T^{2} \) |
| 7 | \( 1 + (-3.33 + 5.77i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + 62.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + (-43.8 - 25.3i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (37.8 - 21.8i)T + (2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (129. + 74.5i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 - 71.8iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 136. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 10.8iT - 2.97e4T^{2} \) |
| 41 | \( 1 + (19.0 - 32.9i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 - 452. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 116.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (45.3 + 78.5i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (476. - 275. i)T + (1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (379. + 218. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (177. - 308. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-522. + 904. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + 790.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-176. - 101. i)T + (2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-327. - 567. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (79.8 - 46.1i)T + (3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 307. iT - 9.12e5T^{2} \) |
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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
−16.57319561234820942533358201475, −15.83560951852199344520202397488, −15.21121000496790519826262727934, −13.14444944427401473096175597952, −10.98890278251317749814153422344, −10.49074949066490597314927289752, −9.152628035020442640521461905725, −7.947460450682345445428151436492, −6.20800178116343622192036483269, −4.48603180814409695098554820666,
0.21298552752718506213037874656, 2.16823903942751996899734313237, 5.83207742858346389161084367815, 7.60770437135783476148710424133, 8.497275275255705604090968775941, 10.41861989483089759454520206316, 11.20596379828587677521560167942, 12.40848436890979750112404502125, 13.34439120771779239360397801762, 15.41256361812474688679170848591