| L(s) = 1 | + (0.415 − 0.909i)2-s + (0.959 + 0.281i)3-s + (−0.654 − 0.755i)4-s + (−1.26 + 0.814i)5-s + (0.654 − 0.755i)6-s + (−0.142 + 0.989i)7-s + (−0.959 + 0.281i)8-s + (0.841 + 0.540i)9-s + (0.214 + 1.49i)10-s + (1.09 + 2.39i)11-s + (−0.415 − 0.909i)12-s + (0.0324 + 0.225i)13-s + (0.841 + 0.540i)14-s + (−1.44 + 0.424i)15-s + (−0.142 + 0.989i)16-s + (−0.344 + 0.397i)17-s + ⋯ |
| L(s) = 1 | + (0.293 − 0.643i)2-s + (0.553 + 0.162i)3-s + (−0.327 − 0.377i)4-s + (−0.566 + 0.364i)5-s + (0.267 − 0.308i)6-s + (−0.0537 + 0.374i)7-s + (−0.339 + 0.0996i)8-s + (0.280 + 0.180i)9-s + (0.0677 + 0.471i)10-s + (0.329 + 0.722i)11-s + (−0.119 − 0.262i)12-s + (0.00900 + 0.0626i)13-s + (0.224 + 0.144i)14-s + (−0.373 + 0.109i)15-s + (−0.0355 + 0.247i)16-s + (−0.0834 + 0.0963i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 - 0.425i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.905 - 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.81607 + 0.405420i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.81607 + 0.405420i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.959 - 0.281i)T \) |
| 7 | \( 1 + (0.142 - 0.989i)T \) |
| 23 | \( 1 + (-4.75 + 0.615i)T \) |
| good | 5 | \( 1 + (1.26 - 0.814i)T + (2.07 - 4.54i)T^{2} \) |
| 11 | \( 1 + (-1.09 - 2.39i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.0324 - 0.225i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (0.344 - 0.397i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (-3.05 - 3.52i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (3.24 - 3.74i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (-1.60 + 0.470i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (-6.89 - 4.43i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (0.0412 - 0.0265i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-4.52 - 1.32i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + 5.00T + 47T^{2} \) |
| 53 | \( 1 + (-0.865 + 6.02i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (0.0271 + 0.188i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-1.19 + 0.350i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (-3.12 + 6.84i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (4.83 - 10.5i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (-4.58 - 5.29i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (1.14 + 7.96i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (1.70 + 1.09i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (7.18 + 2.10i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (2.01 - 1.29i)T + (40.2 - 88.2i)T^{2} \) |
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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
−9.949040842941885628097725820181, −9.466613324544999899853172412202, −8.525420867032099492784156821144, −7.62039574635626622183542099724, −6.77669775165994024928874718246, −5.55398818222795019486286984720, −4.54200470671409877328390901766, −3.62628201756644869723682496820, −2.81467605230033735481403461593, −1.54767810867716490664490538096,
0.801007951336540636095381043301, 2.75374986067122060024823055122, 3.78339675211114409098516411680, 4.57757790620792341544617323055, 5.67520044855360496231753987998, 6.68970269386387620965775760028, 7.50758615365470492608344514441, 8.137595268616235450278639340226, 8.999938942682348068013100511180, 9.603302731883881012963321393322
