| L(s) = 1 | − 238.·5-s + 590.·7-s − 646.·11-s + 1.79e3i·13-s + 4.42e3i·17-s − 169. i·19-s − 1.56e4i·23-s + 4.13e4·25-s − 1.45e4·29-s + 8.10e3·31-s − 1.40e5·35-s + 8.13e4i·37-s − 3.08e4i·41-s − 9.23e4i·43-s − 1.42e5i·47-s + ⋯ |
| L(s) = 1 | − 1.91·5-s + 1.72·7-s − 0.485·11-s + 0.817i·13-s + 0.899i·17-s − 0.0247i·19-s − 1.28i·23-s + 2.64·25-s − 0.595·29-s + 0.272·31-s − 3.28·35-s + 1.60i·37-s − 0.448i·41-s − 1.16i·43-s − 1.37i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.639 - 0.769i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.639 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.8272133092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8272133092\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 238.T + 1.56e4T^{2} \) |
| 7 | \( 1 - 590.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 646.T + 1.77e6T^{2} \) |
| 13 | \( 1 - 1.79e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 4.42e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 169. iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 1.56e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 1.45e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 8.10e3T + 8.87e8T^{2} \) |
| 37 | \( 1 - 8.13e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 3.08e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 9.23e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 1.42e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 1.83e5T + 2.21e10T^{2} \) |
| 59 | \( 1 + 1.83e4T + 4.21e10T^{2} \) |
| 61 | \( 1 - 2.11e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 4.78e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 5.82e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 3.51e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 4.44e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 8.70e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + 5.72e4iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.53e6T + 8.32e11T^{2} \) |
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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
−10.47466311194095761847700451752, −8.673366447812376004899167899954, −8.353341052670840313508456024299, −7.60137482372041055755589771270, −6.77730876092523285123111497035, −5.17882764155420163581715650045, −4.42004664988037423240966048511, −3.75865114844427811885812448689, −2.25772735899983050546997024343, −1.00052121401634539048365281565,
0.20913899361748085031840812497, 1.23893289036780344319219314428, 2.77619013689463276251689266185, 3.88084396851385377944412508194, 4.73136244921114291377562346774, 5.47243382010299012755091775438, 7.25434201239096652770225177954, 7.78090125104714665151169708579, 8.179034848277262485997676713458, 9.271694264073744413533630134726
