| L(s) = 1 | + (0.989 + 0.142i)2-s + (−0.909 + 0.415i)3-s + (0.959 + 0.281i)4-s + (1.08 − 1.95i)5-s + (−0.959 + 0.281i)6-s + (0.378 − 0.588i)7-s + (0.909 + 0.415i)8-s + (0.654 − 0.755i)9-s + (1.34 − 1.78i)10-s + (−0.745 − 5.18i)11-s + (−0.989 + 0.142i)12-s + (−1.92 − 2.99i)13-s + (0.457 − 0.528i)14-s + (−0.171 + 2.22i)15-s + (0.841 + 0.540i)16-s + (0.765 + 2.60i)17-s + ⋯ |
| L(s) = 1 | + (0.699 + 0.100i)2-s + (−0.525 + 0.239i)3-s + (0.479 + 0.140i)4-s + (0.483 − 0.875i)5-s + (−0.391 + 0.115i)6-s + (0.142 − 0.222i)7-s + (0.321 + 0.146i)8-s + (0.218 − 0.251i)9-s + (0.426 − 0.563i)10-s + (−0.224 − 1.56i)11-s + (−0.285 + 0.0410i)12-s + (−0.534 − 0.832i)13-s + (0.122 − 0.141i)14-s + (−0.0441 + 0.575i)15-s + (0.210 + 0.135i)16-s + (0.185 + 0.632i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.75072 - 0.928303i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.75072 - 0.928303i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.989 - 0.142i)T \) |
| 3 | \( 1 + (0.909 - 0.415i)T \) |
| 5 | \( 1 + (-1.08 + 1.95i)T \) |
| 23 | \( 1 + (-0.308 - 4.78i)T \) |
| good | 7 | \( 1 + (-0.378 + 0.588i)T + (-2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (0.745 + 5.18i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (1.92 + 2.99i)T + (-5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-0.765 - 2.60i)T + (-14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (3.73 + 1.09i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-8.22 + 2.41i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-2.66 + 5.82i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-4.34 - 3.76i)T + (5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (1.31 + 1.51i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-4.36 + 1.99i)T + (28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 - 5.47iT - 47T^{2} \) |
| 53 | \( 1 + (-1.05 + 1.64i)T + (-22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (9.17 - 5.89i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-5.71 + 12.5i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (3.44 + 0.495i)T + (64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (0.0479 - 0.333i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (3.65 - 12.4i)T + (-61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (3.98 - 2.55i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-12.2 - 10.6i)T + (11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-3.75 - 8.22i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-5.51 + 4.78i)T + (13.8 - 96.0i)T^{2} \) |
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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
−10.50131596741989460289231785199, −9.615332173891233347377760781196, −8.451692828459489681706156630866, −7.82211545259042145763277400718, −6.29924310809690384186277431417, −5.80089372663713311830488894593, −4.94187462338222078676214496225, −4.01779117330008907272035435701, −2.70409395048610954302247767942, −0.916503682987639454591301867569,
1.91599802663257541247173877565, 2.73498205794778990825085452868, 4.39451906472271662890850782940, 5.02572646228644686511383330542, 6.27887477778143476691720436702, 6.84345722711947631074524616840, 7.56628166468155421720379273651, 9.027214770879701535485830204155, 10.22248397296266459505253958038, 10.44333044534471439811333650702
