L(s) = 1 | + 4i·2-s + (−1.79 − 15.4i)3-s − 16·4-s − 69.6·5-s + (61.9 − 7.18i)6-s − 161. i·7-s − 64i·8-s + (−236. + 55.6i)9-s − 278. i·10-s + 206.·11-s + (28.7 + 247. i)12-s + 54.3·13-s + 646.·14-s + (124. + 1.07e3i)15-s + 256·16-s − 1.07e3·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.115 − 0.993i)3-s − 0.5·4-s − 1.24·5-s + (0.702 − 0.0814i)6-s − 1.24i·7-s − 0.353i·8-s + (−0.973 + 0.228i)9-s − 0.880i·10-s + 0.514·11-s + (0.0575 + 0.496i)12-s + 0.0892·13-s + 0.882·14-s + (0.143 + 1.23i)15-s + 0.250·16-s − 0.906·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.332 - 0.943i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.332 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.3822683248\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3822683248\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 4iT \) |
| 3 | \( 1 + (1.79 + 15.4i)T \) |
| 23 | \( 1 + (-563. - 2.47e3i)T \) |
good | 5 | \( 1 + 69.6T + 3.12e3T^{2} \) |
| 7 | \( 1 + 161. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 206.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 54.3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.07e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 503. iT - 2.47e6T^{2} \) |
| 29 | \( 1 - 7.25e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 4.83e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.90e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 86.4iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 7.00e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 6.59e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 3.21e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.54e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 2.41e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 5.06e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 4.03e3iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 1.08e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.77e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 7.24e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.70e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.65e5iT - 8.58e9T^{2} \) |
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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
−12.72579212424031332122622786679, −11.61020045969966727105197081618, −10.80623428977040365693128202432, −9.059098380775108308268980605428, −7.931708872325872740166533001091, −7.25776550470954522197365717024, −6.46625013714277298205832521866, −4.72132677976876660457206516532, −3.50002150688368481165713079530, −1.08494375906612659083349984814,
0.16322545152581284925129720493, 2.56957049967196799632222064469, 3.86045067339244382932463941745, 4.76678287576579094082129974575, 6.21757580165367580702298952086, 8.203820063581489644942880035554, 8.881887252541898640672182327525, 9.949375394640077571218088086311, 11.20547594504967150032105068080, 11.71973124049153742334410377118