| L(s) = 1 | + (−1 − 1.73i)2-s + (−1.99 + 3.46i)4-s + (5.22 + 9.04i)5-s + (−12.3 − 13.7i)7-s + 7.99·8-s + (10.4 − 18.0i)10-s + (−30.5 + 52.9i)11-s − 59.2·13-s + (−11.4 + 35.2i)14-s + (−8 − 13.8i)16-s + (−10.2 + 17.7i)17-s + (40.1 + 69.5i)19-s − 41.7·20-s + 122.·22-s + (79.1 + 137. i)23-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.467 + 0.809i)5-s + (−0.669 − 0.743i)7-s + 0.353·8-s + (0.330 − 0.572i)10-s + (−0.837 + 1.45i)11-s − 1.26·13-s + (−0.218 + 0.672i)14-s + (−0.125 − 0.216i)16-s + (−0.145 + 0.252i)17-s + (0.485 + 0.840i)19-s − 0.467·20-s + 1.18·22-s + (0.717 + 1.24i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.145 - 0.989i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.145 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.436613 + 0.505542i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.436613 + 0.505542i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1 + 1.73i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (12.3 + 13.7i)T \) |
| good | 5 | \( 1 + (-5.22 - 9.04i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (30.5 - 52.9i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 59.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + (10.2 - 17.7i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-40.1 - 69.5i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-79.1 - 137. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 85.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + (121. - 210. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (145. + 251. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 168T + 6.89e4T^{2} \) |
| 43 | \( 1 - 7.62T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-84.6 - 146. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-125. + 216. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-402. + 697. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (16.5 + 28.7i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-138. + 240. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 631.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-384. + 665. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-209. - 362. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 761.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-786. - 1.36e3i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.04e3T + 9.12e5T^{2} \) |
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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
−12.95937448423676737248158434171, −12.26006799756190123874461406409, −10.78863651547897132630071226956, −10.08554675282715390342660813823, −9.472243527677998301842396325131, −7.56892226174014047775280635041, −6.96796977731882550833358806452, −5.11852399797098339356760306197, −3.45178931008482918989633262652, −2.07297364917298969944364265602,
0.36641668518863912306773905752, 2.72022205158737906360544173768, 5.01973198956340014473638086608, 5.77833682129597488886901496070, 7.14514331579886921539529247145, 8.561596535214405508747673469796, 9.179868164251782194389409565392, 10.24715213598626805680977377477, 11.61801107163965260911149908763, 12.92561439003126717312235273829
