| L(s) = 1 | + 8.47i·5-s + (6.54 + 2.48i)7-s − 13.7·11-s − 0.214i·13-s + 25.4i·17-s + 7.83i·19-s + 40.5·23-s − 46.7·25-s − 21.8·29-s − 14.3i·31-s + (−21.0 + 55.4i)35-s − 40.3·37-s − 14.1i·41-s + 28.9·43-s + 32.9i·47-s + ⋯ |
| L(s) = 1 | + 1.69i·5-s + (0.935 + 0.354i)7-s − 1.24·11-s − 0.0165i·13-s + 1.49i·17-s + 0.412i·19-s + 1.76·23-s − 1.86·25-s − 0.753·29-s − 0.462i·31-s + (−0.600 + 1.58i)35-s − 1.08·37-s − 0.344i·41-s + 0.672·43-s + 0.700i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 - 0.354i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.935 - 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.403626236\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.403626236\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-6.54 - 2.48i)T \) |
| good | 5 | \( 1 - 8.47iT - 25T^{2} \) |
| 11 | \( 1 + 13.7T + 121T^{2} \) |
| 13 | \( 1 + 0.214iT - 169T^{2} \) |
| 17 | \( 1 - 25.4iT - 289T^{2} \) |
| 19 | \( 1 - 7.83iT - 361T^{2} \) |
| 23 | \( 1 - 40.5T + 529T^{2} \) |
| 29 | \( 1 + 21.8T + 841T^{2} \) |
| 31 | \( 1 + 14.3iT - 961T^{2} \) |
| 37 | \( 1 + 40.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 14.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 28.9T + 1.84e3T^{2} \) |
| 47 | \( 1 - 32.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 56.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + 96.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 103. iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 45.9T + 4.48e3T^{2} \) |
| 71 | \( 1 + 41.1T + 5.04e3T^{2} \) |
| 73 | \( 1 - 134. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 9.69T + 6.24e3T^{2} \) |
| 83 | \( 1 + 14.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 58.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 8.14iT - 9.40e3T^{2} \) |
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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
−10.51119112635632378922889503818, −9.391381371882373285975936815888, −8.220834982851543715042867084920, −7.68825800223528851237866019309, −6.81722663155723006305054777609, −5.89292344800877340555464840995, −5.05298682988541240502501585714, −3.71630797370715133196690436519, −2.77968124729492574771297487347, −1.83375304257198647595862985407,
0.43579238292894566998343605602, 1.47468120233580653781498527267, 2.87050742773028783213075845867, 4.44229760400895726342722517834, 5.03045837387126493791637973987, 5.48681932423059736569783574465, 7.19922445471585902698723018847, 7.72103870647875378906666072395, 8.791143336643594368228337274883, 9.062626716186043726535149381376
