| L(s) = 1 | + (−0.173 + 0.984i)2-s + (1.13 + 1.30i)3-s + (−0.939 − 0.342i)4-s + (2.14 − 1.80i)5-s + (−1.48 + 0.888i)6-s + (0.939 − 0.342i)7-s + (0.5 − 0.866i)8-s + (−0.431 + 2.96i)9-s + (1.40 + 2.42i)10-s + (0.617 + 0.518i)11-s + (−0.616 − 1.61i)12-s + (0.0751 + 0.426i)13-s + (0.173 + 0.984i)14-s + (4.79 + 0.770i)15-s + (0.766 + 0.642i)16-s + (0.882 + 1.52i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 + 0.696i)2-s + (0.654 + 0.756i)3-s + (−0.469 − 0.171i)4-s + (0.960 − 0.805i)5-s + (−0.606 + 0.362i)6-s + (0.355 − 0.129i)7-s + (0.176 − 0.306i)8-s + (−0.143 + 0.989i)9-s + (0.443 + 0.767i)10-s + (0.186 + 0.156i)11-s + (−0.178 − 0.467i)12-s + (0.0208 + 0.118i)13-s + (0.0464 + 0.263i)14-s + (1.23 + 0.199i)15-s + (0.191 + 0.160i)16-s + (0.214 + 0.370i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.46726 + 1.01349i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.46726 + 1.01349i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.173 - 0.984i)T \) |
| 3 | \( 1 + (-1.13 - 1.30i)T \) |
| 7 | \( 1 + (-0.939 + 0.342i)T \) |
| good | 5 | \( 1 + (-2.14 + 1.80i)T + (0.868 - 4.92i)T^{2} \) |
| 11 | \( 1 + (-0.617 - 0.518i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.0751 - 0.426i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.882 - 1.52i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.43 + 4.21i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.01 - 0.732i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (1.34 - 7.62i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (9.12 + 3.32i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (0.701 + 1.21i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.0980 + 0.555i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (7.28 + 6.11i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (7.86 - 2.86i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + 0.683T + 53T^{2} \) |
| 59 | \( 1 + (-10.6 + 8.94i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.529 + 0.192i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (2.22 + 12.6i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-6.31 - 10.9i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.627 + 1.08i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.10 + 6.24i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.124 + 0.706i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-3.20 + 5.55i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.94 - 8.34i)T + (16.8 + 95.5i)T^{2} \) |
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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
−11.31042547005981963295839660232, −10.32298928591086102819702976816, −9.310841024134347693603111944458, −9.033015843591540659960216722392, −7.979204888902321195154191095179, −6.89418938135816991706024834389, −5.39563634632768388010381792660, −4.97010314674145875818895720423, −3.59597397228426283793004159630, −1.77878015263461027412526537842,
1.55048298118435140760264826061, 2.60054035046269373194608980504, 3.63785791346384066531404415163, 5.44484153316138694315063492946, 6.48810369927598806155100043076, 7.54227354335347183679767425136, 8.487428309088505310804469605631, 9.519134190520134238821129783703, 10.13631727286394769161599199464, 11.27551790018441628261697987618
