| L(s) = 1 | + (−0.0393 + 0.273i)2-s + (0.955 + 2.09i)3-s + (1.84 + 0.541i)4-s + (−0.654 − 0.755i)5-s + (−0.610 + 0.179i)6-s + (−2.63 − 1.69i)7-s + (−0.450 + 0.986i)8-s + (−1.50 + 1.73i)9-s + (0.232 − 0.149i)10-s + (−0.369 − 2.57i)11-s + (0.629 + 4.38i)12-s + (−0.0559 + 0.0359i)13-s + (0.567 − 0.654i)14-s + (0.955 − 2.09i)15-s + (2.98 + 1.91i)16-s + (−1.42 + 0.417i)17-s + ⋯ |
| L(s) = 1 | + (−0.0278 + 0.193i)2-s + (0.551 + 1.20i)3-s + (0.922 + 0.270i)4-s + (−0.292 − 0.337i)5-s + (−0.249 + 0.0731i)6-s + (−0.997 − 0.640i)7-s + (−0.159 + 0.348i)8-s + (−0.500 + 0.577i)9-s + (0.0735 − 0.0472i)10-s + (−0.111 − 0.775i)11-s + (0.181 + 1.26i)12-s + (−0.0155 + 0.00997i)13-s + (0.151 − 0.175i)14-s + (0.246 − 0.540i)15-s + (0.746 + 0.479i)16-s + (−0.344 + 0.101i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 - 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.496 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.09082 + 0.632321i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09082 + 0.632321i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.654 + 0.755i)T \) |
| 23 | \( 1 + (2.65 + 3.99i)T \) |
| good | 2 | \( 1 + (0.0393 - 0.273i)T + (-1.91 - 0.563i)T^{2} \) |
| 3 | \( 1 + (-0.955 - 2.09i)T + (-1.96 + 2.26i)T^{2} \) |
| 7 | \( 1 + (2.63 + 1.69i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (0.369 + 2.57i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (0.0559 - 0.0359i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (1.42 - 0.417i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-4.42 - 1.30i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (9.93 - 2.91i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-3.45 + 7.56i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (3.92 - 4.53i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (2.55 + 2.95i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-3.26 - 7.15i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 8.42T + 47T^{2} \) |
| 53 | \( 1 + (0.190 + 0.122i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-0.709 + 0.455i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (3.25 - 7.12i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-1.42 + 9.89i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (1.22 - 8.51i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (8.56 + 2.51i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (12.5 - 8.04i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (5.87 - 6.78i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-0.984 - 2.15i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-5.06 - 5.84i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−13.89322124314557443874777037379, −12.74516186580684436482523872203, −11.48464931885178393967156559787, −10.49127634545959453786198422226, −9.579033172756370034221662341900, −8.433601597350122034700944057847, −7.23094951123732915281684949102, −5.85335034310328917411313136136, −4.03969650581240791419955337778, −3.08760014670368757284715812525,
2.01359399109839184178184501764, 3.18144092403093910052359326337, 5.82387441288195187302251020657, 7.02053917112974513955514243071, 7.51232369873892292541188361660, 9.129922423515619609889475977588, 10.25496100430016167565520995100, 11.66267734609107290084109577552, 12.33958473968082900153718751573, 13.22300708356734936102813814683
