| L(s) = 1 | + (−15.4 + 1.94i)3-s + 101.·5-s − 200. i·7-s + (235. − 60.0i)9-s + 350. i·11-s + 744. i·13-s + (−1.56e3 + 196. i)15-s + 1.44e3i·17-s + 1.31e3·19-s + (389. + 3.09e3i)21-s − 2.04e3·23-s + 7.09e3·25-s + (−3.52e3 + 1.38e3i)27-s − 3.78e3·29-s + 1.42e3i·31-s + ⋯ |
| L(s) = 1 | + (−0.992 + 0.124i)3-s + 1.80·5-s − 1.54i·7-s + (0.968 − 0.247i)9-s + 0.872i·11-s + 1.22i·13-s + (−1.79 + 0.225i)15-s + 1.21i·17-s + 0.834·19-s + (0.192 + 1.53i)21-s − 0.804·23-s + 2.27·25-s + (−0.930 + 0.366i)27-s − 0.835·29-s + 0.266i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.613 - 0.789i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.613 - 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.028651612\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.028651612\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (15.4 - 1.94i)T \) |
| good | 5 | \( 1 - 101.T + 3.12e3T^{2} \) |
| 7 | \( 1 + 200. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 350. iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 744. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.44e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.31e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.04e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.78e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.42e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 2.61e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.08e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 1.61e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 3.41e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 539.T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.37e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 5.04e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 1.57e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.64e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.70e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.32e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 7.58e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 9.73e3iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 1.15e5T + 8.58e9T^{2} \) |
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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
−10.43738960506382101093490352855, −9.957532536344575602501041377164, −9.285634462400263380797784214855, −7.50277312509154130633477550385, −6.65484283162660269392528603243, −5.98186533186089191536993266993, −4.84587897943442299781605220002, −3.96225111666324316931434710146, −1.88844454288170948174299148826, −1.18507693355874128312060666372,
0.60913713957265843181553040461, 1.95117951338896693711291765018, 2.92477341655085752187074922605, 5.10597774894203368374702690289, 5.79835956684369847334320744582, 5.93957946403785020332106151619, 7.40663288740626442181773282515, 8.830806100881973501591070703989, 9.562684911484902956898665616409, 10.32972959642428045852404973353
