| L(s) = 1 | + (−0.896 + 1.09i)2-s + (−0.456 − 0.683i)3-s + (−0.390 − 1.96i)4-s + (−1.61 − 0.322i)5-s + (1.15 + 0.113i)6-s + (4.21 − 0.837i)7-s + (2.49 + 1.33i)8-s + (0.889 − 2.14i)9-s + (1.80 − 1.48i)10-s + (−0.106 − 0.0712i)11-s + (−1.16 + 1.16i)12-s + (2.31 − 2.31i)13-s + (−2.86 + 5.35i)14-s + (0.518 + 1.25i)15-s + (−3.69 + 1.53i)16-s + (−2.89 − 2.93i)17-s + ⋯ |
| L(s) = 1 | + (−0.634 + 0.773i)2-s + (−0.263 − 0.394i)3-s + (−0.195 − 0.980i)4-s + (−0.723 − 0.144i)5-s + (0.472 + 0.0464i)6-s + (1.59 − 0.316i)7-s + (0.882 + 0.470i)8-s + (0.296 − 0.715i)9-s + (0.570 − 0.468i)10-s + (−0.0321 − 0.0214i)11-s + (−0.335 + 0.335i)12-s + (0.642 − 0.642i)13-s + (−0.764 + 1.43i)14-s + (0.134 + 0.323i)15-s + (−0.923 + 0.383i)16-s + (−0.702 − 0.711i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.754046 - 0.102019i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.754046 - 0.102019i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.896 - 1.09i)T \) |
| 17 | \( 1 + (2.89 + 2.93i)T \) |
| good | 3 | \( 1 + (0.456 + 0.683i)T + (-1.14 + 2.77i)T^{2} \) |
| 5 | \( 1 + (1.61 + 0.322i)T + (4.61 + 1.91i)T^{2} \) |
| 7 | \( 1 + (-4.21 + 0.837i)T + (6.46 - 2.67i)T^{2} \) |
| 11 | \( 1 + (0.106 + 0.0712i)T + (4.20 + 10.1i)T^{2} \) |
| 13 | \( 1 + (-2.31 + 2.31i)T - 13iT^{2} \) |
| 19 | \( 1 + (-1.68 - 4.06i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-2.14 - 1.43i)T + (8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (-0.908 + 4.56i)T + (-26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 + (-2.02 - 3.02i)T + (-11.8 + 28.6i)T^{2} \) |
| 37 | \( 1 + (7.00 - 4.68i)T + (14.1 - 34.1i)T^{2} \) |
| 41 | \( 1 + (-2.02 - 10.1i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (-3.26 + 7.87i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (4.90 - 4.90i)T - 47iT^{2} \) |
| 53 | \( 1 + (-1.32 + 0.549i)T + (37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-12.6 - 5.25i)T + (41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (-0.813 - 4.09i)T + (-56.3 + 23.3i)T^{2} \) |
| 67 | \( 1 - 3.72iT - 67T^{2} \) |
| 71 | \( 1 + (0.759 - 0.507i)T + (27.1 - 65.5i)T^{2} \) |
| 73 | \( 1 + (0.631 - 3.17i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (6.49 - 9.71i)T + (-30.2 - 72.9i)T^{2} \) |
| 83 | \( 1 + (4.75 - 1.97i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (0.886 - 0.886i)T - 89iT^{2} \) |
| 97 | \( 1 + (8.67 + 1.72i)T + (89.6 + 37.1i)T^{2} \) |
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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
−13.38658206788293277756646147308, −11.85517306859397372628183448537, −11.21809200222550826411533558262, −10.01337383435287615868779027750, −8.565472310850821473516660432892, −7.88463349992905277965384408937, −6.93623373182657868862230389531, −5.54361629998077734205676267018, −4.27512985700063491299114506052, −1.19190008027357791088342298393,
1.94886893179146205391740649507, 4.01214100740954046967226976369, 4.98154371654707423880543603169, 7.21033007303717695719705130357, 8.211810592667792272002556750213, 9.008419484999063228081677666457, 10.57879868518173760817063069877, 11.18160661872807058351359010749, 11.72312465695666079654986646934, 13.05135602137385918265844842470
