| L(s) = 1 | − 2.49·3-s + 2.23i·5-s − 1.88i·7-s − 2.76·9-s + 16.3i·11-s + 15.4·13-s − 5.58i·15-s − 12.7i·17-s + 34.9i·19-s + 4.71i·21-s + (16.4 − 16.0i)23-s − 5.00·25-s + 29.3·27-s + 8.12·29-s − 53.1·31-s + ⋯ |
| L(s) = 1 | − 0.832·3-s + 0.447i·5-s − 0.269i·7-s − 0.306·9-s + 1.48i·11-s + 1.19·13-s − 0.372i·15-s − 0.749i·17-s + 1.83i·19-s + 0.224i·21-s + (0.716 − 0.697i)23-s − 0.200·25-s + 1.08·27-s + 0.280·29-s − 1.71·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.697 - 0.716i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.697 - 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9126577523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9126577523\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (-16.4 + 16.0i)T \) |
| good | 3 | \( 1 + 2.49T + 9T^{2} \) |
| 7 | \( 1 + 1.88iT - 49T^{2} \) |
| 11 | \( 1 - 16.3iT - 121T^{2} \) |
| 13 | \( 1 - 15.4T + 169T^{2} \) |
| 17 | \( 1 + 12.7iT - 289T^{2} \) |
| 19 | \( 1 - 34.9iT - 361T^{2} \) |
| 29 | \( 1 - 8.12T + 841T^{2} \) |
| 31 | \( 1 + 53.1T + 961T^{2} \) |
| 37 | \( 1 + 30.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 63.4T + 1.68e3T^{2} \) |
| 43 | \( 1 + 3.36iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 31.3T + 2.20e3T^{2} \) |
| 53 | \( 1 - 31.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 27.0T + 3.48e3T^{2} \) |
| 61 | \( 1 - 31.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 28.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 38.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + 45.1T + 5.32e3T^{2} \) |
| 79 | \( 1 - 87.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 73.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 12.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 35.0iT - 9.40e3T^{2} \) |
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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
−9.441493600447985754848322788249, −8.607343548619250932164383178316, −7.55700108126624050963281452063, −7.00114381349457318671170296719, −6.06360951932302785646192544077, −5.50956454183595341779079785945, −4.44819893251391821689437175796, −3.62768164286095367955870324283, −2.38331588266375537951665064248, −1.16357703658954418985386302724,
0.31667159576191298113863513098, 1.25397032182434546952992358154, 2.84286990179170809918400655623, 3.72672074572354962821872990611, 4.90231528097431812949322842475, 5.66847210446622113881387921929, 6.11834762538691991513723303545, 7.01768960639958137024654198895, 8.227230421001278110201657830089, 8.781798559796600215943535850096
