| L(s) = 1 | + (0.309 + 0.951i)3-s − 5-s + (−1.93 − 1.40i)7-s + (−0.809 + 0.587i)9-s + (−1.81 − 1.31i)11-s + (1.72 + 5.32i)13-s + (−0.309 − 0.951i)15-s + (0.108 − 0.0787i)17-s + (−0.515 + 1.58i)19-s + (0.740 − 2.27i)21-s + (3.73 − 2.71i)23-s + 25-s + (−0.809 − 0.587i)27-s + (0.277 − 0.855i)29-s + (−3.54 − 4.29i)31-s + ⋯ |
| L(s) = 1 | + (0.178 + 0.549i)3-s − 0.447·5-s + (−0.732 − 0.532i)7-s + (−0.269 + 0.195i)9-s + (−0.546 − 0.396i)11-s + (0.479 + 1.47i)13-s + (−0.0797 − 0.245i)15-s + (0.0262 − 0.0190i)17-s + (−0.118 + 0.364i)19-s + (0.161 − 0.497i)21-s + (0.778 − 0.565i)23-s + 0.200·25-s + (−0.155 − 0.113i)27-s + (0.0516 − 0.158i)29-s + (−0.637 − 0.770i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.211 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6096544552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6096544552\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.309 - 0.951i)T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 + (3.54 + 4.29i)T \) |
| good | 7 | \( 1 + (1.93 + 1.40i)T + (2.16 + 6.65i)T^{2} \) |
| 11 | \( 1 + (1.81 + 1.31i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-1.72 - 5.32i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.108 + 0.0787i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (0.515 - 1.58i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-3.73 + 2.71i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.277 + 0.855i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + 7.79T + 37T^{2} \) |
| 41 | \( 1 + (-0.441 + 1.35i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-3.14 + 9.68i)T + (-34.7 - 25.2i)T^{2} \) |
| 47 | \( 1 + (3.28 + 10.1i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-6.90 + 5.01i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (3.96 + 12.1i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 - 0.692T + 67T^{2} \) |
| 71 | \( 1 + (9.29 - 6.75i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (11.7 + 8.50i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-3.41 + 2.48i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (4.88 - 15.0i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-8.78 - 6.38i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (12.7 + 9.22i)T + (29.9 + 92.2i)T^{2} \) |
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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
−8.916229947784420031164228714546, −8.484638819528244955632515360679, −7.32900571510914552072380452746, −6.76262560903778733511766970730, −5.79148605199009209979239679226, −4.77689075398345842103982125127, −3.88246382075403952871284672770, −3.33966311217863845557450105438, −2.01378935746724745466426094647, −0.22557294643068909253343607296,
1.29520171750144236340069561482, 2.87966100160715988485122663496, 3.18475564865343805042456513208, 4.60757010957834475897470330065, 5.58843440757069769544130469724, 6.23760951429064655158858765664, 7.32244696109379499389959005561, 7.72412981605053259054635502388, 8.725800529545071662589582673623, 9.225723868716761688418434230112
