| L(s) = 1 | + (−0.665 + 1.30i)2-s + (−0.130 − 0.822i)3-s + (−0.0875 − 0.120i)4-s + (−1.11 + 1.93i)5-s + (1.16 + 0.377i)6-s + (4.16 + 0.659i)7-s + (−2.68 + 0.424i)8-s + (2.19 − 0.712i)9-s + (−1.78 − 2.74i)10-s + (−0.0877 + 0.0877i)12-s + (0.824 + 0.420i)13-s + (−3.63 + 4.99i)14-s + (1.73 + 0.667i)15-s + (1.32 − 4.06i)16-s + (−1.71 + 0.875i)17-s + (−0.529 + 3.34i)18-s + ⋯ |
| L(s) = 1 | + (−0.470 + 0.923i)2-s + (−0.0751 − 0.474i)3-s + (−0.0437 − 0.0602i)4-s + (−0.500 + 0.865i)5-s + (0.473 + 0.153i)6-s + (1.57 + 0.249i)7-s + (−0.947 + 0.150i)8-s + (0.731 − 0.237i)9-s + (−0.564 − 0.869i)10-s + (−0.0253 + 0.0253i)12-s + (0.228 + 0.116i)13-s + (−0.970 + 1.33i)14-s + (0.448 + 0.172i)15-s + (0.330 − 1.01i)16-s + (−0.416 + 0.212i)17-s + (−0.124 + 0.787i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.432 - 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.432 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.678059 + 1.07764i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.678059 + 1.07764i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.11 - 1.93i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (0.665 - 1.30i)T + (-1.17 - 1.61i)T^{2} \) |
| 3 | \( 1 + (0.130 + 0.822i)T + (-2.85 + 0.927i)T^{2} \) |
| 7 | \( 1 + (-4.16 - 0.659i)T + (6.65 + 2.16i)T^{2} \) |
| 13 | \( 1 + (-0.824 - 0.420i)T + (7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (1.71 - 0.875i)T + (9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (-4.39 - 3.19i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.95 - 1.95i)T + 23iT^{2} \) |
| 29 | \( 1 + (0.810 - 0.588i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.131 - 0.403i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (0.771 - 4.87i)T + (-35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (0.339 - 0.467i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (5.05 - 5.05i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.17 - 0.186i)T + (44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (4.12 - 8.09i)T + (-31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (-5.47 - 7.53i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (7.40 + 2.40i)T + (49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (3.05 - 3.05i)T - 67iT^{2} \) |
| 71 | \( 1 + (-2.65 + 8.17i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.854 + 5.39i)T + (-69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (-0.705 - 2.17i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (0.902 + 1.77i)T + (-48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + 13.9iT - 89T^{2} \) |
| 97 | \( 1 + (2.96 + 1.50i)T + (57.0 + 78.4i)T^{2} \) |
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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
−11.09706172638375341189967463462, −9.972776394033108944676769630889, −8.824101354566364554809906818445, −7.912504310461188413450882345714, −7.54515944219997385699364645542, −6.72390847985672668256909466843, −5.81167633898494080331436476820, −4.53668864873197865585399444693, −3.18005693394479937877710811186, −1.62417075790690138671610039838,
0.937951512139612548630237377841, 1.98993413317545271486839587868, 3.67072988267911419412140599017, 4.72768114317971611280759242741, 5.32755629859516343598779965034, 7.03153642798392192034769923645, 8.005583701181067665725371611213, 8.837283789347618195353897932300, 9.578694216871723516223146223691, 10.53536670234532910797720187459
