L(s) = 1 | + (−7.95 + 4.20i)3-s + 19.3i·5-s + 67.6·7-s + (45.6 − 66.9i)9-s − 77.9i·11-s − 79.2·13-s + (−81.2 − 153. i)15-s − 309. i·17-s − 468.·19-s + (−538. + 284. i)21-s + 617. i·23-s + 251.·25-s + (−82.1 + 724. i)27-s − 939. i·29-s − 1.10e3·31-s + ⋯ |
L(s) = 1 | + (−0.884 + 0.467i)3-s + 0.773i·5-s + 1.38·7-s + (0.563 − 0.825i)9-s − 0.644i·11-s − 0.468·13-s + (−0.361 − 0.683i)15-s − 1.07i·17-s − 1.29·19-s + (−1.22 + 0.644i)21-s + 1.16i·23-s + 0.401·25-s + (−0.112 + 0.993i)27-s − 1.11i·29-s − 1.14·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.467 + 0.884i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.467 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.086929265\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.086929265\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (7.95 - 4.20i)T \) |
good | 5 | \( 1 - 19.3iT - 625T^{2} \) |
| 7 | \( 1 - 67.6T + 2.40e3T^{2} \) |
| 11 | \( 1 + 77.9iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 79.2T + 2.85e4T^{2} \) |
| 17 | \( 1 + 309. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 468.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 617. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 939. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.10e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 720.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 2.30e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 2.03e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 2.15e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 2.37e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 2.58e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 3.13e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 2.47e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 7.74e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 1.02e4T + 2.83e7T^{2} \) |
| 79 | \( 1 - 5.31e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 1.06e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 2.02e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.68e4T + 8.85e7T^{2} \) |
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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
−10.79490495654751765270996817448, −9.874651941794505996544896810674, −8.765543980513746479010553087833, −7.60585713250728525989951872425, −6.72181588077429478538532265846, −5.56874555479237712132773972048, −4.79048820174667044649701216701, −3.61213892265329505925805045697, −2.02360834998366908052067788433, −0.37360425790386157686043567818,
1.19896975834138343945315419714, 2.06252416898407955368989584682, 4.46467820111705780441720304716, 4.84566106160496793143389872596, 6.01090726445333921983203636878, 7.10645391081458767418573827556, 8.099710583715357466314614833489, 8.790541776441622949755568135606, 10.27504974234931678377610957561, 10.91910876850479213022701554118