| L(s) = 1 | + (−1.02 + 2.25i)2-s + (2.16 + 0.636i)3-s + (−2.71 − 3.13i)4-s + (0.841 − 0.540i)5-s + (−3.66 + 4.23i)6-s + (−0.617 + 4.29i)7-s + (5.10 − 1.50i)8-s + (1.77 + 1.13i)9-s + (0.352 + 2.45i)10-s + (−1.51 − 3.32i)11-s + (−3.89 − 8.52i)12-s + (−0.268 − 1.86i)13-s + (−9.05 − 5.81i)14-s + (2.16 − 0.636i)15-s + (−0.698 + 4.85i)16-s + (3.37 − 3.89i)17-s + ⋯ |
| L(s) = 1 | + (−0.728 + 1.59i)2-s + (1.25 + 0.367i)3-s + (−1.35 − 1.56i)4-s + (0.376 − 0.241i)5-s + (−1.49 + 1.72i)6-s + (−0.233 + 1.62i)7-s + (1.80 − 0.530i)8-s + (0.590 + 0.379i)9-s + (0.111 + 0.776i)10-s + (−0.457 − 1.00i)11-s + (−1.12 − 2.46i)12-s + (−0.0745 − 0.518i)13-s + (−2.41 − 1.55i)14-s + (0.559 − 0.164i)15-s + (−0.174 + 1.21i)16-s + (0.818 − 0.944i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.442209 + 0.896776i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.442209 + 0.896776i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.841 + 0.540i)T \) |
| 23 | \( 1 + (4.69 - 0.957i)T \) |
| good | 2 | \( 1 + (1.02 - 2.25i)T + (-1.30 - 1.51i)T^{2} \) |
| 3 | \( 1 + (-2.16 - 0.636i)T + (2.52 + 1.62i)T^{2} \) |
| 7 | \( 1 + (0.617 - 4.29i)T + (-6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (1.51 + 3.32i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (0.268 + 1.86i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (-3.37 + 3.89i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (-3.25 - 3.75i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (-3.85 + 4.45i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (1.82 - 0.536i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (-4.20 - 2.69i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (2.74 - 1.76i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-0.333 - 0.0978i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + 1.89T + 47T^{2} \) |
| 53 | \( 1 + (0.408 - 2.83i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (2.01 + 14.0i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (2.79 - 0.820i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (2.01 - 4.41i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (3.32 - 7.27i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (5.80 + 6.70i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (-0.523 - 3.64i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-12.4 - 7.99i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (4.64 + 1.36i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (11.3 - 7.28i)T + (40.2 - 88.2i)T^{2} \) |
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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
−14.27222230067016657879619394412, −13.57313445573983602451064539615, −12.01548889546811367140257845790, −9.862405218815966403888362979631, −9.392697247067484724745493129254, −8.336915573983114571752352915557, −7.947135062066858925803331514156, −6.05032705307321991875898145998, −5.35243724389690720802492424337, −2.93938633242375520086569217368,
1.65392973687874324476951196128, 3.02313950684859309301896809163, 4.17213416021298015244344017668, 7.17393166867706506741170343203, 8.015316870806489525197260164788, 9.282845972193105604964725912558, 10.07624267746844393486777266036, 10.75581486753632635829571057608, 12.20941411366733540336118099376, 13.20028591928954229945721884268
