| L(s) = 1 | + (11.0 − 11.0i)3-s + (75.4 − 99.6i)5-s + (337. + 337. i)7-s − 242. i·9-s − 1.14e3·11-s + (2.56e3 − 2.56e3i)13-s + (−267. − 1.93e3i)15-s + (837. + 837. i)17-s + 238. i·19-s + 7.43e3·21-s + (1.15e4 − 1.15e4i)23-s + (−4.24e3 − 1.50e4i)25-s + (−2.67e3 − 2.67e3i)27-s − 2.26e4i·29-s + 2.02e4·31-s + ⋯ |
| L(s) = 1 | + (0.408 − 0.408i)3-s + (0.603 − 0.797i)5-s + (0.983 + 0.983i)7-s − 0.333i·9-s − 0.861·11-s + (1.16 − 1.16i)13-s + (−0.0792 − 0.571i)15-s + (0.170 + 0.170i)17-s + 0.0347i·19-s + 0.802·21-s + (0.948 − 0.948i)23-s + (−0.271 − 0.962i)25-s + (−0.136 − 0.136i)27-s − 0.927i·29-s + 0.679·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.25723 - 1.06203i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.25723 - 1.06203i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-11.0 + 11.0i)T \) |
| 5 | \( 1 + (-75.4 + 99.6i)T \) |
| good | 7 | \( 1 + (-337. - 337. i)T + 1.17e5iT^{2} \) |
| 11 | \( 1 + 1.14e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + (-2.56e3 + 2.56e3i)T - 4.82e6iT^{2} \) |
| 17 | \( 1 + (-837. - 837. i)T + 2.41e7iT^{2} \) |
| 19 | \( 1 - 238. iT - 4.70e7T^{2} \) |
| 23 | \( 1 + (-1.15e4 + 1.15e4i)T - 1.48e8iT^{2} \) |
| 29 | \( 1 + 2.26e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 2.02e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + (-4.41e4 - 4.41e4i)T + 2.56e9iT^{2} \) |
| 41 | \( 1 + 6.30e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + (8.02e4 - 8.02e4i)T - 6.32e9iT^{2} \) |
| 47 | \( 1 + (7.91e4 + 7.91e4i)T + 1.07e10iT^{2} \) |
| 53 | \( 1 + (8.22e4 - 8.22e4i)T - 2.21e10iT^{2} \) |
| 59 | \( 1 - 3.81e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 1.68e4T + 5.15e10T^{2} \) |
| 67 | \( 1 + (-2.76e5 - 2.76e5i)T + 9.04e10iT^{2} \) |
| 71 | \( 1 - 2.89e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (3.74e5 - 3.74e5i)T - 1.51e11iT^{2} \) |
| 79 | \( 1 - 6.85e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + (-3.71e5 + 3.71e5i)T - 3.26e11iT^{2} \) |
| 89 | \( 1 + 1.02e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (9.28e5 + 9.28e5i)T + 8.32e11iT^{2} \) |
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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
−13.44358418804081664568792122651, −12.82542885348871449678836567829, −11.55796714226770668436943723362, −10.13167683644564428175501180480, −8.556703551027660920991940144651, −8.164888600632220136807962153534, −6.03477220668880577181125861172, −4.95001396712414661789305447509, −2.64828257946832102349011130451, −1.16023472591203926429150816714,
1.67515725024649028700515045289, 3.47007977456713225570389687381, 5.01981954585919702204284949424, 6.77865220312941972422437775494, 8.007744470099176344504829356433, 9.432975894673219269640984818455, 10.69194513477694842618349259158, 11.26010399757318795267480000310, 13.38248014287384971022633012057, 13.97414317321665121635546304680
