L(s) = 1 | + (1.20 − 0.742i)2-s + (0.707 − 0.707i)3-s + (0.898 − 1.78i)4-s + (0.860 + 2.06i)5-s + (0.326 − 1.37i)6-s − 0.707·7-s + (−0.244 − 2.81i)8-s − 1.00i·9-s + (2.56 + 1.84i)10-s + (1.79 − 1.79i)11-s + (−0.628 − 1.89i)12-s + (−3.86 + 3.86i)13-s + (−0.851 + 0.524i)14-s + (2.06 + 0.850i)15-s + (−2.38 − 3.21i)16-s + 0.244i·17-s + ⋯ |
L(s) = 1 | + (0.851 − 0.524i)2-s + (0.408 − 0.408i)3-s + (0.449 − 0.893i)4-s + (0.384 + 0.922i)5-s + (0.133 − 0.561i)6-s − 0.267·7-s + (−0.0863 − 0.996i)8-s − 0.333i·9-s + (0.812 + 0.583i)10-s + (0.542 − 0.542i)11-s + (−0.181 − 0.548i)12-s + (−1.07 + 1.07i)13-s + (−0.227 + 0.140i)14-s + (0.533 + 0.219i)15-s + (−0.596 − 0.802i)16-s + 0.0593i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.598 + 0.801i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.598 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.95818 - 0.981648i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.95818 - 0.981648i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.20 + 0.742i)T \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (-0.860 - 2.06i)T \) |
good | 7 | \( 1 + 0.707T + 7T^{2} \) |
| 11 | \( 1 + (-1.79 + 1.79i)T - 11iT^{2} \) |
| 13 | \( 1 + (3.86 - 3.86i)T - 13iT^{2} \) |
| 17 | \( 1 - 0.244iT - 17T^{2} \) |
| 19 | \( 1 + (-1.53 - 1.53i)T + 19iT^{2} \) |
| 23 | \( 1 + 6.92T + 23T^{2} \) |
| 29 | \( 1 + (4.89 + 4.89i)T + 29iT^{2} \) |
| 31 | \( 1 - 7.60T + 31T^{2} \) |
| 37 | \( 1 + (-8.47 - 8.47i)T + 37iT^{2} \) |
| 41 | \( 1 - 2.12iT - 41T^{2} \) |
| 43 | \( 1 + (-0.684 - 0.684i)T + 43iT^{2} \) |
| 47 | \( 1 - 4.47iT - 47T^{2} \) |
| 53 | \( 1 + (1.47 + 1.47i)T + 53iT^{2} \) |
| 59 | \( 1 + (-5.86 + 5.86i)T - 59iT^{2} \) |
| 61 | \( 1 + (-0.0537 - 0.0537i)T + 61iT^{2} \) |
| 67 | \( 1 + (-7.85 + 7.85i)T - 67iT^{2} \) |
| 71 | \( 1 + 2.08iT - 71T^{2} \) |
| 73 | \( 1 + 9.69T + 73T^{2} \) |
| 79 | \( 1 + 7.34T + 79T^{2} \) |
| 83 | \( 1 + (-6.80 + 6.80i)T - 83iT^{2} \) |
| 89 | \( 1 + 3.07iT - 89T^{2} \) |
| 97 | \( 1 - 1.39iT - 97T^{2} \) |
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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
−11.84870479957560637257502551978, −11.43898740238839909997633570974, −9.963426691601558360073673241155, −9.577324193967937897604825534149, −7.81763723707092330610099211055, −6.61802705399806848001133188289, −6.04129664314537904920120998604, −4.33826980465850332137698030332, −3.10392869833740293905464813253, −1.97647857477971439053557347247,
2.43876316184202812763620288744, 3.96075727441017176208927438165, 4.96946005159928655091882681518, 5.88703730405084705608471539798, 7.30305660168177527491317502248, 8.218284956892367780633333101623, 9.326849385626968390840312438271, 10.16582863126688907044232218822, 11.69633177584465970704693970983, 12.54316854076634477583151658302