L(s) = 1 | + (−0.826 + 0.563i)2-s + (0.365 − 0.930i)4-s + (−0.365 − 0.930i)7-s + (0.222 + 0.974i)8-s + (0.0747 − 0.997i)9-s + (0.109 + 1.46i)11-s + (−0.900 − 0.433i)13-s + (0.826 + 0.563i)14-s + (−0.733 − 0.680i)16-s + (−1.63 − 0.246i)17-s + (0.5 + 0.866i)18-s + (0.0747 − 0.129i)19-s + (−0.914 − 1.14i)22-s + (0.826 + 0.563i)25-s + (0.988 − 0.149i)26-s + ⋯ |
L(s) = 1 | + (−0.826 + 0.563i)2-s + (0.365 − 0.930i)4-s + (−0.365 − 0.930i)7-s + (0.222 + 0.974i)8-s + (0.0747 − 0.997i)9-s + (0.109 + 1.46i)11-s + (−0.900 − 0.433i)13-s + (0.826 + 0.563i)14-s + (−0.733 − 0.680i)16-s + (−1.63 − 0.246i)17-s + (0.5 + 0.866i)18-s + (0.0747 − 0.129i)19-s + (−0.914 − 1.14i)22-s + (0.826 + 0.563i)25-s + (0.988 − 0.149i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.304 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.304 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3989276386\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3989276386\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.826 - 0.563i)T \) |
| 7 | \( 1 + (0.365 + 0.930i)T \) |
| 13 | \( 1 + (0.900 + 0.433i)T \) |
good | 3 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 5 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 11 | \( 1 + (-0.109 - 1.46i)T + (-0.988 + 0.149i)T^{2} \) |
| 17 | \( 1 + (1.63 + 0.246i)T + (0.955 + 0.294i)T^{2} \) |
| 19 | \( 1 + (-0.0747 + 0.129i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 29 | \( 1 + (-1.19 + 1.49i)T + (-0.222 - 0.974i)T^{2} \) |
| 31 | \( 1 + (0.900 + 1.56i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 41 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 43 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 47 | \( 1 + (1.57 - 1.07i)T + (0.365 - 0.930i)T^{2} \) |
| 53 | \( 1 + (0.365 - 0.930i)T + (-0.733 - 0.680i)T^{2} \) |
| 59 | \( 1 + (0.698 - 0.215i)T + (0.826 - 0.563i)T^{2} \) |
| 61 | \( 1 + (0.365 + 0.930i)T + (-0.733 + 0.680i)T^{2} \) |
| 67 | \( 1 + (0.988 + 1.71i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (1.03 + 1.29i)T + (-0.222 + 0.974i)T^{2} \) |
| 73 | \( 1 + (-0.365 - 0.930i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-1.12 + 0.541i)T + (0.623 - 0.781i)T^{2} \) |
| 89 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.208473984812952177201293208253, −7.80649974890830825813950603373, −7.44759394465055014578215380637, −6.59955771430853612321336837292, −6.24862773781326545348192040341, −4.75486255014698663597513241731, −4.40301438212980905864951612452, −2.91324341145270040845542747667, −1.80208442575299469267476657626, −0.32732673443606019708119674068,
1.64212138597743798653548030918, 2.61970004669131806992988833852, 3.24351441504279358667770621491, 4.55138656650202942187605861975, 5.35119639367540302841538910889, 6.58043438225980940952312266605, 6.95786576487147179803209341790, 8.184216178863053965903348778052, 8.751202292713443950762873515473, 8.992744637470260723363354276647