L(s) = 1 | + (−0.414 − 2.79i)2-s + (−7.65 + 2.31i)4-s + 8.74i·5-s + (−13.8 + 12.3i)7-s + (9.65 + 20.4i)8-s + (24.4 − 3.62i)10-s − 0.397i·11-s − 75.7i·13-s + (40.2 + 33.4i)14-s + (53.2 − 35.4i)16-s − 84.4i·17-s + 20.0·19-s + (−20.2 − 66.9i)20-s + (−1.11 + 0.164i)22-s − 122. i·23-s + ⋯ |
L(s) = 1 | + (−0.146 − 0.989i)2-s + (−0.957 + 0.289i)4-s + 0.782i·5-s + (−0.745 + 0.666i)7-s + (0.426 + 0.904i)8-s + (0.773 − 0.114i)10-s − 0.0109i·11-s − 1.61i·13-s + (0.768 + 0.639i)14-s + (0.832 − 0.554i)16-s − 1.20i·17-s + 0.242·19-s + (−0.226 − 0.748i)20-s + (−0.0107 + 0.00159i)22-s − 1.10i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.519 + 0.854i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.519 + 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.020160294\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.020160294\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.414 + 2.79i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (13.8 - 12.3i)T \) |
good | 5 | \( 1 - 8.74iT - 125T^{2} \) |
| 11 | \( 1 + 0.397iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 75.7iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 84.4iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 20.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 122. iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 175.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 128.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 253.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 81.4iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 69.1iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 147.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 283.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 632.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 3.25iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 551. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 486. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 165. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 279. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 622.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 696. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 1.62e3iT - 9.12e5T^{2} \) |
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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
−11.15838049921616950251079897906, −10.39084392636513060329895433580, −9.603792035913856867572470914366, −8.592262420898629994713513056576, −7.45926836797596625531971045565, −6.09166446244299667084121204862, −4.87983717727416780062713648079, −3.17435395130982011735776171050, −2.65975018653638702929702114511, −0.48990779076566087795949744242,
1.24084448970809512818125974727, 3.80765250014432627294164915546, 4.70502370528893507951738886305, 6.05010910718309676323554715708, 6.88915496543217349459998875721, 7.944062025704278679215674814919, 9.044074389801328591620605665531, 9.608304938169377126183280429581, 10.76318480695355258497993865976, 12.17823230410806252477840970595