| L(s) = 1 | + (−0.391 − 1.20i)2-s + (−1.46 − 2.31i)3-s + (0.317 − 0.230i)4-s + (1.07 − 2.28i)5-s + (−2.21 + 2.67i)6-s + (2.55 + 2.39i)7-s + (−2.45 − 1.78i)8-s + (−1.92 + 4.08i)9-s + (−3.18 − 0.401i)10-s + (0.328 − 0.517i)11-s + (−1.00 − 0.396i)12-s + (−0.283 + 4.50i)13-s + (1.88 − 4.01i)14-s + (−6.87 + 0.868i)15-s + (−0.945 + 2.91i)16-s + (−2.79 + 2.62i)17-s + ⋯ |
| L(s) = 1 | + (−0.277 − 0.852i)2-s + (−0.847 − 1.33i)3-s + (0.158 − 0.115i)4-s + (0.481 − 1.02i)5-s + (−0.904 + 1.09i)6-s + (0.963 + 0.905i)7-s + (−0.867 − 0.630i)8-s + (−0.640 + 1.36i)9-s + (−1.00 − 0.127i)10-s + (0.0990 − 0.156i)11-s + (−0.288 − 0.114i)12-s + (−0.0785 + 1.24i)13-s + (0.504 − 1.07i)14-s + (−1.77 + 0.224i)15-s + (−0.236 + 0.727i)16-s + (−0.678 + 0.637i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.147474 - 0.907561i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.147474 - 0.907561i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 151 | \( 1 + (-5.27 + 11.0i)T \) |
| good | 2 | \( 1 + (0.391 + 1.20i)T + (-1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (1.46 + 2.31i)T + (-1.27 + 2.71i)T^{2} \) |
| 5 | \( 1 + (-1.07 + 2.28i)T + (-3.18 - 3.85i)T^{2} \) |
| 7 | \( 1 + (-2.55 - 2.39i)T + (0.439 + 6.98i)T^{2} \) |
| 11 | \( 1 + (-0.328 + 0.517i)T + (-4.68 - 9.95i)T^{2} \) |
| 13 | \( 1 + (0.283 - 4.50i)T + (-12.8 - 1.62i)T^{2} \) |
| 17 | \( 1 + (2.79 - 2.62i)T + (1.06 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-6.36 + 4.62i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (0.214 - 0.156i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-8.78 - 2.25i)T + (25.4 + 13.9i)T^{2} \) |
| 31 | \( 1 + (7.99 + 4.39i)T + (16.6 + 26.1i)T^{2} \) |
| 37 | \( 1 + (-0.0827 + 1.31i)T + (-36.7 - 4.63i)T^{2} \) |
| 41 | \( 1 + (0.625 - 0.756i)T + (-7.68 - 40.2i)T^{2} \) |
| 43 | \( 1 + (1.69 - 1.58i)T + (2.69 - 42.9i)T^{2} \) |
| 47 | \( 1 + (-5.57 + 6.74i)T + (-8.80 - 46.1i)T^{2} \) |
| 53 | \( 1 + (12.1 - 4.79i)T + (38.6 - 36.2i)T^{2} \) |
| 59 | \( 1 + (-0.708 + 2.17i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (0.612 - 0.964i)T + (-25.9 - 55.1i)T^{2} \) |
| 67 | \( 1 + (-2.91 - 6.19i)T + (-42.7 + 51.6i)T^{2} \) |
| 71 | \( 1 + (2.37 + 2.22i)T + (4.45 + 70.8i)T^{2} \) |
| 73 | \( 1 + (-4.74 - 4.45i)T + (4.58 + 72.8i)T^{2} \) |
| 79 | \( 1 + (3.38 - 1.85i)T + (42.3 - 66.7i)T^{2} \) |
| 83 | \( 1 + (-1.23 + 6.46i)T + (-77.1 - 30.5i)T^{2} \) |
| 89 | \( 1 + (2.03 - 10.6i)T + (-82.7 - 32.7i)T^{2} \) |
| 97 | \( 1 + (5.20 + 11.0i)T + (-61.8 + 74.7i)T^{2} \) |
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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
−12.28380321464962151633673783722, −11.62605436560741485525655940363, −11.09145532580761150263407263135, −9.394903519303317494230063076100, −8.626790829730291813056306798008, −7.07501111527265723349491499520, −5.98891343689138156363035778780, −4.99639084800285245787200239721, −2.15280274204727542667045224927, −1.25818495449340258992441831748,
3.23736261220210848106357862481, 4.88681911895857826515228248253, 5.86276785445473720602821295953, 7.04009909636339536040459427722, 8.023619533674067086147376792785, 9.602158686577309134888211932442, 10.53713211257044182442182466301, 11.05401870044079683059807146413, 12.08118810883501952167610829907, 14.03669734700442559672054091117
