| L(s) = 1 | + (−0.936 − 0.349i)2-s + (1.26 + 0.691i)3-s + (0.755 + 0.654i)4-s + (−0.584 − 2.15i)5-s + (−0.944 − 1.09i)6-s + (1.35 − 1.80i)7-s + (−0.479 − 0.877i)8-s + (−0.496 − 0.773i)9-s + (−0.206 + 2.22i)10-s + (−0.475 − 0.217i)11-s + (0.504 + 1.35i)12-s + (3.11 − 2.32i)13-s + (−1.90 + 1.22i)14-s + (0.751 − 3.13i)15-s + (0.142 + 0.989i)16-s + (−0.578 + 8.08i)17-s + ⋯ |
| L(s) = 1 | + (−0.662 − 0.247i)2-s + (0.730 + 0.399i)3-s + (0.377 + 0.327i)4-s + (−0.261 − 0.965i)5-s + (−0.385 − 0.445i)6-s + (0.511 − 0.683i)7-s + (−0.169 − 0.310i)8-s + (−0.165 − 0.257i)9-s + (−0.0652 + 0.704i)10-s + (−0.143 − 0.0654i)11-s + (0.145 + 0.390i)12-s + (0.863 − 0.646i)13-s + (−0.508 + 0.326i)14-s + (0.194 − 0.809i)15-s + (0.0355 + 0.247i)16-s + (−0.140 + 1.96i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.648 + 0.761i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.648 + 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.02806 - 0.474848i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02806 - 0.474848i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.936 + 0.349i)T \) |
| 5 | \( 1 + (0.584 + 2.15i)T \) |
| 23 | \( 1 + (0.430 - 4.77i)T \) |
| good | 3 | \( 1 + (-1.26 - 0.691i)T + (1.62 + 2.52i)T^{2} \) |
| 7 | \( 1 + (-1.35 + 1.80i)T + (-1.97 - 6.71i)T^{2} \) |
| 11 | \( 1 + (0.475 + 0.217i)T + (7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (-3.11 + 2.32i)T + (3.66 - 12.4i)T^{2} \) |
| 17 | \( 1 + (0.578 - 8.08i)T + (-16.8 - 2.41i)T^{2} \) |
| 19 | \( 1 + (-4.90 + 5.66i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (-1.93 + 1.67i)T + (4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (0.910 + 0.267i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (-2.51 - 0.546i)T + (33.6 + 15.3i)T^{2} \) |
| 41 | \( 1 + (5.19 + 3.34i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (4.84 - 8.86i)T + (-23.2 - 36.1i)T^{2} \) |
| 47 | \( 1 + (0.388 - 0.388i)T - 47iT^{2} \) |
| 53 | \( 1 + (-7.32 - 5.48i)T + (14.9 + 50.8i)T^{2} \) |
| 59 | \( 1 + (7.55 + 1.08i)T + (56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (3.43 - 11.6i)T + (-51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (3.30 - 8.86i)T + (-50.6 - 43.8i)T^{2} \) |
| 71 | \( 1 + (2.35 + 5.15i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-0.283 + 0.0202i)T + (72.2 - 10.3i)T^{2} \) |
| 79 | \( 1 + (-0.529 + 3.68i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (1.46 - 6.74i)T + (-75.4 - 34.4i)T^{2} \) |
| 89 | \( 1 + (-9.07 + 2.66i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (1.22 + 5.61i)T + (-88.2 + 40.2i)T^{2} \) |
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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
−11.89083041284411398996024678999, −11.00625688710225646745296972940, −10.00934543391799523897547439709, −8.989168484775841761039141605994, −8.355801179115253818283666941495, −7.57603411841554202820194970319, −5.88971785767565487351266742195, −4.32729294557142124243976012675, −3.29673488039680282973564673581, −1.24253687831595956381206300499,
2.05725886595382197113136654700, 3.21111053515755037131443139486, 5.23070748482315376716246744596, 6.60130715649703933855735180148, 7.53721656427392872138169615206, 8.311355181512590429198490937537, 9.203548708419353482233873335165, 10.33724966753981248071795403162, 11.40345280972902021651537785746, 11.98323046525244300615702932474
