| L(s) = 1 | + (−0.415 + 0.909i)2-s + (0.959 + 0.281i)3-s + (−0.654 − 0.755i)4-s + (0.520 − 0.334i)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 + (0.0880 + 0.612i)10-s + (−1.93 − 4.24i)11-s + (−0.415 − 0.909i)12-s + (−0.977 − 6.79i)13-s + (−0.841 − 0.540i)14-s + (0.593 − 0.174i)15-s + (−0.142 + 0.989i)16-s + (2.58 − 2.98i)17-s + ⋯ |
| L(s) = 1 | + (−0.293 + 0.643i)2-s + (0.553 + 0.162i)3-s + (−0.327 − 0.377i)4-s + (0.232 − 0.149i)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 + (0.0278 + 0.193i)10-s + (−0.584 − 1.27i)11-s + (−0.119 − 0.262i)12-s + (−0.271 − 1.88i)13-s + (−0.224 − 0.144i)14-s + (0.153 − 0.0449i)15-s + (−0.0355 + 0.247i)16-s + (0.626 − 0.723i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.720 + 0.693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.720 + 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.20158 - 0.484323i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20158 - 0.484323i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.27 + 4.22i)T \) |
| good | 5 | \( 1 + (-0.520 + 0.334i)T + (2.07 - 4.54i)T^{2} \) |
| 11 | \( 1 + (1.93 + 4.24i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (0.977 + 6.79i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (-2.58 + 2.98i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (3.65 + 4.22i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (4.82 - 5.56i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (-5.66 + 1.66i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (-0.741 - 0.476i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (7.43 - 4.78i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-11.6 - 3.41i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 - 0.617T + 47T^{2} \) |
| 53 | \( 1 + (-1.06 + 7.41i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (0.479 + 3.33i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-3.03 + 0.892i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (-3.34 + 7.33i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (-5.47 + 11.9i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (-1.86 - 2.15i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (-2.09 - 14.5i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-0.128 - 0.0827i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (5.21 + 1.53i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (-10.2 + 6.61i)T + (40.2 - 88.2i)T^{2} \) |
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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
−9.761016079670511164529537535471, −8.942365682950479598128823642126, −8.174638275984965367976259205938, −7.72178740995116419832517531821, −6.48774624484933914582636120063, −5.54189010932199753073556946102, −4.97760967608866831948882863682, −3.40974735936235487944424455193, −2.56113437135134942154761135671, −0.60739682142223050914055093640,
1.75271673476159277030974269857, 2.31551997289354046459962594208, 3.92204592225791412488939640655, 4.35122245170670637466596175472, 5.88656144070210838357601102149, 6.98844617333905760554290943996, 7.69170976348093921374037252517, 8.522169749775565792252771433109, 9.565496580270894606023695823409, 9.966880154119231466200046084266
