| L(s) = 1 | − 15.5·3-s − 20i·5-s + 155. i·7-s + 243·9-s + 2.30e3·11-s − 3.22e3i·13-s + 311. i·15-s − 6.56e3·17-s − 3.92e3·19-s − 2.42e3i·21-s − 1.73e4i·23-s + 1.52e4·25-s − 3.78e3·27-s + 4.47e4i·29-s + 3.06e4i·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.160i·5-s + 0.454i·7-s + 0.333·9-s + 1.73·11-s − 1.46i·13-s + 0.0923i·15-s − 1.33·17-s − 0.572·19-s − 0.262i·21-s − 1.42i·23-s + 0.974·25-s − 0.192·27-s + 1.83i·29-s + 1.02i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.376982000\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.376982000\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 15.5T \) |
| good | 5 | \( 1 + 20iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 155. iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 2.30e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + 3.22e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 6.56e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + 3.92e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + 1.73e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 4.47e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 3.06e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 4.41e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 9.00e3T + 4.75e9T^{2} \) |
| 43 | \( 1 - 2.52e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 1.75e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 9.62e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 3.50e4T + 4.21e10T^{2} \) |
| 61 | \( 1 - 8.61e4iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 4.24e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 3.65e4iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 3.75e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 5.20e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 5.94e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 9.01e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + 1.25e6T + 8.32e11T^{2} \) |
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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
−10.36356391665281005333171523353, −8.907445107242139961148303307030, −8.618875974490035822627063524399, −6.94912318608254140082771552312, −6.40927454649185228932212856172, −5.23854381555457996418841804620, −4.31415106560083646913645760454, −3.00419726790973500676537927225, −1.52985905572160712526736098560, −0.38863684617602233642247960013,
1.03708052945737353333269523231, 2.13833767920018886120978310854, 4.02946147548982452888783385065, 4.37339809120159559225412284775, 6.08219388297588756835561899267, 6.63432101858278833602374639082, 7.53176304350010356443692668415, 9.086721761533736628681032081128, 9.425657194705329142798769508908, 10.80791334542613846594307409151
