L(s) = 1 | + (−0.157 + 1.95i)2-s + (−0.845 − 0.534i)3-s + (−1.83 − 0.297i)4-s + (−2.06 + 2.32i)5-s + (1.17 − 1.56i)6-s + (−0.626 − 0.601i)7-s + (−0.0680 + 0.276i)8-s + (0.428 + 0.903i)9-s + (−4.23 − 4.40i)10-s + (1.11 + 0.531i)11-s + (1.38 + 1.23i)12-s + (−3.59 − 0.238i)13-s + (1.27 − 1.13i)14-s + (2.98 − 0.865i)15-s + (−4.04 − 1.35i)16-s + (2.79 − 2.91i)17-s + ⋯ |
L(s) = 1 | + (−0.111 + 1.38i)2-s + (−0.487 − 0.308i)3-s + (−0.915 − 0.148i)4-s + (−0.922 + 1.04i)5-s + (0.481 − 0.640i)6-s + (−0.236 − 0.227i)7-s + (−0.0240 + 0.0976i)8-s + (0.142 + 0.301i)9-s + (−1.33 − 1.39i)10-s + (0.337 + 0.160i)11-s + (0.400 + 0.355i)12-s + (−0.997 − 0.0661i)13-s + (0.341 − 0.302i)14-s + (0.771 − 0.223i)15-s + (−1.01 − 0.337i)16-s + (0.678 − 0.706i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0997 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0997 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.117707 - 0.106495i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.117707 - 0.106495i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.845 + 0.534i)T \) |
| 13 | \( 1 + (3.59 + 0.238i)T \) |
good | 2 | \( 1 + (0.157 - 1.95i)T + (-1.97 - 0.320i)T^{2} \) |
| 5 | \( 1 + (2.06 - 2.32i)T + (-0.602 - 4.96i)T^{2} \) |
| 7 | \( 1 + (0.626 + 0.601i)T + (0.281 + 6.99i)T^{2} \) |
| 11 | \( 1 + (-1.11 - 0.531i)T + (6.95 + 8.52i)T^{2} \) |
| 17 | \( 1 + (-2.79 + 2.91i)T + (-0.684 - 16.9i)T^{2} \) |
| 19 | \( 1 + (2.44 + 1.41i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.19 + 2.06i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.53 + 0.204i)T + (28.6 + 4.65i)T^{2} \) |
| 31 | \( 1 + (-1.31 - 0.159i)T + (30.0 + 7.41i)T^{2} \) |
| 37 | \( 1 + (2.78 + 6.54i)T + (-25.6 + 26.6i)T^{2} \) |
| 41 | \( 1 + (3.02 - 4.78i)T + (-17.5 - 37.0i)T^{2} \) |
| 43 | \( 1 + (2.78 + 1.18i)T + (29.7 + 31.0i)T^{2} \) |
| 47 | \( 1 + (-4.83 - 1.83i)T + (35.1 + 31.1i)T^{2} \) |
| 53 | \( 1 + (9.34 + 2.30i)T + (46.9 + 24.6i)T^{2} \) |
| 59 | \( 1 + (-4.77 - 14.2i)T + (-47.1 + 35.4i)T^{2} \) |
| 61 | \( 1 + (3.99 - 13.7i)T + (-51.5 - 32.6i)T^{2} \) |
| 67 | \( 1 + (0.0467 + 0.287i)T + (-63.5 + 21.2i)T^{2} \) |
| 71 | \( 1 + (7.55 + 0.304i)T + (70.7 + 5.71i)T^{2} \) |
| 73 | \( 1 + (-1.45 + 1.00i)T + (25.8 - 68.2i)T^{2} \) |
| 79 | \( 1 + (1.23 - 3.25i)T + (-59.1 - 52.3i)T^{2} \) |
| 83 | \( 1 + (6.34 - 12.0i)T + (-47.1 - 68.3i)T^{2} \) |
| 89 | \( 1 + (5.23 - 3.02i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.09 + 1.03i)T + (89.2 - 38.0i)T^{2} \) |
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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
−11.62947084325026456060377205457, −10.73937314085764463720027065992, −9.750806197680672645148093583068, −8.532716844684074645010423084392, −7.43477938391469292816059845436, −7.21666810910799402339310287225, −6.40136994454571087698599570067, −5.33899167913433591455906105034, −4.20368468713715228930824442153, −2.70265577057971793980556869897,
0.10291277029948207658978148449, 1.61994857205256095178668465005, 3.31383011693233905666168913362, 4.18041072257922140367460259107, 5.08004295312170288484543740416, 6.41057637049554434600573998019, 7.78318263819666023803515733251, 8.762712773705725751336621660486, 9.606873871476448931171787972427, 10.31352188911507631395843215730