L(s) = 1 | + (0.628 − 0.362i)2-s + (−2.99 − 4.24i)3-s + (−3.73 + 6.47i)4-s + (−5.53 + 9.58i)5-s + (−3.42 − 1.58i)6-s + (13.3 + 12.8i)7-s + 11.2i·8-s + (−9.10 + 25.4i)9-s + 8.03i·10-s + (0.219 − 0.126i)11-s + (38.6 − 3.48i)12-s + (−12.3 − 7.15i)13-s + (13.0 + 3.26i)14-s + (57.2 − 5.15i)15-s + (−25.8 − 44.7i)16-s − 92.5·17-s + ⋯ |
L(s) = 1 | + (0.222 − 0.128i)2-s + (−0.575 − 0.817i)3-s + (−0.467 + 0.808i)4-s + (−0.494 + 0.857i)5-s + (−0.232 − 0.107i)6-s + (0.718 + 0.695i)7-s + 0.496i·8-s + (−0.337 + 0.941i)9-s + 0.253i·10-s + (0.00602 − 0.00347i)11-s + (0.930 − 0.0838i)12-s + (−0.264 − 0.152i)13-s + (0.248 + 0.0622i)14-s + (0.985 − 0.0888i)15-s + (−0.403 − 0.698i)16-s − 1.32·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.108 - 0.994i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.108 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.571628 + 0.637621i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.571628 + 0.637621i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.99 + 4.24i)T \) |
| 7 | \( 1 + (-13.3 - 12.8i)T \) |
good | 2 | \( 1 + (-0.628 + 0.362i)T + (4 - 6.92i)T^{2} \) |
| 5 | \( 1 + (5.53 - 9.58i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-0.219 + 0.126i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (12.3 + 7.15i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + 92.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 130. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (102. + 59.3i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-248. + 143. i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-214. - 123. i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 188.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-53.2 + 92.2i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (21.5 + 37.3i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (68.8 + 119. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 - 419. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (217. - 376. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (163. - 94.3i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (185. - 321. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 26.1iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 728. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (48.6 + 84.3i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-401. - 695. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 - 236.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-1.26e3 + 732. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−14.50771773438695815534921053021, −13.59201069582938645703440218884, −12.20901407161870533117400495073, −11.81835626729349954581797106150, −10.58126984988852280866923524265, −8.467192268500192449491771811721, −7.68418421331663797051986448931, −6.24174294241584239964760718714, −4.53766116399526029551649261324, −2.53873494886484665176150758521,
0.60389137709429538346149519275, 4.46707313524555258545941163498, 4.78611321938994744768295621753, 6.52503197550332287638316442977, 8.493766051597122905495033360621, 9.599754432407758726274725071124, 10.77923490901380440474515665066, 11.71158130722884320415326234126, 13.19638471097445331260278833990, 14.27734359000304234348589926847