| L(s) = 1 | + (−0.415 − 0.909i)2-s + (0.959 − 0.281i)3-s + (−0.654 + 0.755i)4-s + (2.47 + 1.59i)5-s + (−0.654 − 0.755i)6-s + (−0.151 − 1.05i)7-s + (0.959 + 0.281i)8-s + (0.841 − 0.540i)9-s + (0.418 − 2.91i)10-s + (−1.74 + 3.81i)11-s + (−0.415 + 0.909i)12-s + (0.834 − 5.80i)13-s + (−0.897 + 0.577i)14-s + (2.82 + 0.829i)15-s + (−0.142 − 0.989i)16-s + (−3.51 − 4.05i)17-s + ⋯ |
| L(s) = 1 | + (−0.293 − 0.643i)2-s + (0.553 − 0.162i)3-s + (−0.327 + 0.377i)4-s + (1.10 + 0.711i)5-s + (−0.267 − 0.308i)6-s + (−0.0574 − 0.399i)7-s + (0.339 + 0.0996i)8-s + (0.280 − 0.180i)9-s + (0.132 − 0.921i)10-s + (−0.525 + 1.15i)11-s + (−0.119 + 0.262i)12-s + (0.231 − 1.61i)13-s + (−0.239 + 0.154i)14-s + (0.729 + 0.214i)15-s + (−0.0355 − 0.247i)16-s + (−0.852 − 0.983i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.808 + 0.588i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.808 + 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.15110 - 0.374549i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15110 - 0.374549i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.415 + 0.909i)T \) |
| 3 | \( 1 + (-0.959 + 0.281i)T \) |
| 23 | \( 1 + (2.13 - 4.29i)T \) |
| good | 5 | \( 1 + (-2.47 - 1.59i)T + (2.07 + 4.54i)T^{2} \) |
| 7 | \( 1 + (0.151 + 1.05i)T + (-6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (1.74 - 3.81i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.834 + 5.80i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (3.51 + 4.05i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (1.59 - 1.83i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (-5.90 - 6.81i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (4.40 + 1.29i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (7.22 - 4.64i)T + (15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (3.67 + 2.36i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (-0.521 + 0.153i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + 1.61T + 47T^{2} \) |
| 53 | \( 1 + (0.409 + 2.85i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (0.0614 - 0.427i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (-1.04 - 0.305i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (1.19 + 2.61i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (2.66 + 5.84i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-8.77 + 10.1i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (-0.186 + 1.29i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (-10.9 + 7.04i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (6.91 - 2.02i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (-9.65 - 6.20i)T + (40.2 + 88.2i)T^{2} \) |
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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
−13.21515881729677278480511327309, −12.20444662076609807485318196302, −10.55205723519454258847883366516, −10.23832376826310709095162068029, −9.172520646402697861668337067693, −7.83031202006060426397337819589, −6.79067674326340582779382572243, −5.14800121239298898439100220478, −3.25689785232330629896753700182, −2.05273822401174831161255440278,
2.08353848498447627827694367798, 4.34449728126390379446370564303, 5.73004335874429933020551492015, 6.64641650442286581230791700134, 8.533784469398340921980083195095, 8.775924836597778170713504168897, 9.867733737207894325332505656838, 11.02099733374377158371465446957, 12.61190019227986104975922845898, 13.66347181549954610845835860860
