| L(s) = 1 | + (−1.26 − 0.642i)2-s + (−1.71 − 0.270i)3-s + (1.17 + 1.61i)4-s + (−4.55 + 2.05i)5-s + (1.98 + 1.43i)6-s + (3.78 − 3.78i)7-s + (−0.442 − 2.79i)8-s + (2.85 + 0.927i)9-s + (7.06 + 0.336i)10-s + (3.18 + 9.81i)11-s + (−1.57 − 3.08i)12-s + (15.1 − 7.70i)13-s + (−7.19 + 2.33i)14-s + (8.35 − 2.28i)15-s + (−1.23 + 3.80i)16-s + (21.2 − 3.36i)17-s + ⋯ |
| L(s) = 1 | + (−0.630 − 0.321i)2-s + (−0.570 − 0.0903i)3-s + (0.293 + 0.404i)4-s + (−0.911 + 0.411i)5-s + (0.330 + 0.239i)6-s + (0.540 − 0.540i)7-s + (−0.0553 − 0.349i)8-s + (0.317 + 0.103i)9-s + (0.706 + 0.0336i)10-s + (0.289 + 0.892i)11-s + (−0.131 − 0.257i)12-s + (1.16 − 0.592i)13-s + (−0.514 + 0.167i)14-s + (0.556 − 0.152i)15-s + (−0.0772 + 0.237i)16-s + (1.24 − 0.197i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.339i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.940 + 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.850261 - 0.148805i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.850261 - 0.148805i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.26 + 0.642i)T \) |
| 3 | \( 1 + (1.71 + 0.270i)T \) |
| 5 | \( 1 + (4.55 - 2.05i)T \) |
| good | 7 | \( 1 + (-3.78 + 3.78i)T - 49iT^{2} \) |
| 11 | \( 1 + (-3.18 - 9.81i)T + (-97.8 + 71.1i)T^{2} \) |
| 13 | \( 1 + (-15.1 + 7.70i)T + (99.3 - 136. i)T^{2} \) |
| 17 | \( 1 + (-21.2 + 3.36i)T + (274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (-5.71 + 7.86i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (-3.73 + 7.32i)T + (-310. - 427. i)T^{2} \) |
| 29 | \( 1 + (-0.300 - 0.414i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (-27.9 - 20.2i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (-27.7 - 54.4i)T + (-804. + 1.10e3i)T^{2} \) |
| 41 | \( 1 + (6.25 - 19.2i)T + (-1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (53.4 + 53.4i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-2.72 + 17.2i)T + (-2.10e3 - 682. i)T^{2} \) |
| 53 | \( 1 + (26.4 + 4.19i)T + (2.67e3 + 868. i)T^{2} \) |
| 59 | \( 1 + (48.8 + 15.8i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-27.3 - 84.0i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + (-68.3 + 10.8i)T + (4.26e3 - 1.38e3i)T^{2} \) |
| 71 | \( 1 + (-100. + 73.2i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-8.59 + 16.8i)T + (-3.13e3 - 4.31e3i)T^{2} \) |
| 79 | \( 1 + (50.3 + 69.3i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (5.02 + 31.7i)T + (-6.55e3 + 2.12e3i)T^{2} \) |
| 89 | \( 1 + (-144. + 47.0i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (1.86 - 11.7i)T + (-8.94e3 - 2.90e3i)T^{2} \) |
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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
−12.29682808557735953554945730833, −11.62546482327393990091939396299, −10.73919785772643390515577813693, −9.969579821882971505574333754587, −8.376207457357858833324796124889, −7.55890779838808099433021413142, −6.54944160054394498691275697215, −4.75443836419312332832205475003, −3.36384335239096288501257219497, −1.06547908643214914341956008715,
1.10470607220416390222243115896, 3.74449990536473104617849468351, 5.30985172717533819156870710947, 6.33145564112894711232609532108, 7.83212735353467223925286699900, 8.510059531450305106111081402430, 9.627762258504093103284118823494, 11.13253171720797440723771815172, 11.49290175397232466649361637003, 12.53559076109969600057756526586
