L(s) = 1 | + (−0.349 + 0.936i)3-s + (0.989 − 0.142i)5-s + (−0.599 + 0.800i)7-s + (−0.755 − 0.654i)9-s + (−0.142 + 0.989i)11-s + (0.936 + 0.349i)13-s + (−0.212 + 0.977i)15-s + (0.540 − 0.841i)17-s + (−0.0713 − 0.997i)19-s + (−0.540 − 0.841i)21-s + (0.997 − 0.0713i)23-s + (0.959 − 0.281i)25-s + (0.877 − 0.479i)27-s + (0.800 + 0.599i)29-s + (0.997 + 0.0713i)31-s + ⋯ |
L(s) = 1 | + (−0.349 + 0.936i)3-s + (0.989 − 0.142i)5-s + (−0.599 + 0.800i)7-s + (−0.755 − 0.654i)9-s + (−0.142 + 0.989i)11-s + (0.936 + 0.349i)13-s + (−0.212 + 0.977i)15-s + (0.540 − 0.841i)17-s + (−0.0713 − 0.997i)19-s + (−0.540 − 0.841i)21-s + (0.997 − 0.0713i)23-s + (0.959 − 0.281i)25-s + (0.877 − 0.479i)27-s + (0.800 + 0.599i)29-s + (0.997 + 0.0713i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0853 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0853 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.697771100 + 1.558595384i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.697771100 + 1.558595384i\) |
\(L(1)\) |
\(\approx\) |
\(1.106075448 + 0.4690104252i\) |
\(L(1)\) |
\(\approx\) |
\(1.106075448 + 0.4690104252i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 89 | \( 1 \) |
good | 3 | \( 1 + (-0.349 + 0.936i)T \) |
| 5 | \( 1 + (0.989 - 0.142i)T \) |
| 7 | \( 1 + (-0.599 + 0.800i)T \) |
| 11 | \( 1 + (-0.142 + 0.989i)T \) |
| 13 | \( 1 + (0.936 + 0.349i)T \) |
| 17 | \( 1 + (0.540 - 0.841i)T \) |
| 19 | \( 1 + (-0.0713 - 0.997i)T \) |
| 23 | \( 1 + (0.997 - 0.0713i)T \) |
| 29 | \( 1 + (0.800 + 0.599i)T \) |
| 31 | \( 1 + (0.997 + 0.0713i)T \) |
| 37 | \( 1 + (-0.707 + 0.707i)T \) |
| 41 | \( 1 + (0.936 - 0.349i)T \) |
| 43 | \( 1 + (0.800 - 0.599i)T \) |
| 47 | \( 1 + (-0.909 - 0.415i)T \) |
| 53 | \( 1 + (-0.909 + 0.415i)T \) |
| 59 | \( 1 + (0.349 + 0.936i)T \) |
| 61 | \( 1 + (0.479 + 0.877i)T \) |
| 67 | \( 1 + (-0.415 - 0.909i)T \) |
| 71 | \( 1 + (0.989 + 0.142i)T \) |
| 73 | \( 1 + (0.654 + 0.755i)T \) |
| 79 | \( 1 + (0.755 - 0.654i)T \) |
| 83 | \( 1 + (-0.212 - 0.977i)T \) |
| 97 | \( 1 + (-0.142 - 0.989i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−22.48010647515020946622930972624, −21.22690012457646200180506710145, −20.80441222877468405169958581112, −19.34713338117482919211758658266, −19.10217899698173652182469870186, −18.07293790796118361739113122229, −17.33615456743683512869103097091, −16.704175312718758824141042647557, −15.90540134600507634674552400565, −14.34387184021894746101283400171, −13.8732174374609377946633468201, −13.04619781513107985516322520067, −12.586206450796283326625819572880, −11.15789586105625040251745388413, −10.633938694697607669728170507586, −9.71081453778786085335768072023, −8.46387023371351766446743434119, −7.77120908442754126415117747619, −6.370803753838317214191525656486, −6.26698113918732643719213989861, −5.23598271586062794737339531778, −3.64524923758522035740363712126, −2.75450570301773044250067299594, −1.40869731587151507949390030349, −0.73338338657165958756876343406,
0.93799121036070266921324365622, 2.43614759059036270533277504849, 3.21210726723649500512544548245, 4.65602939025034406992198776916, 5.22725992119884424377509571286, 6.2050147346306213232835443223, 6.92428484968828627960342994198, 8.68458311104436793013189019339, 9.21074638303437833412349893777, 9.91690319994966875926635354341, 10.71194044488490127884515523062, 11.758284212480901479759777700205, 12.577477677869159260950488662988, 13.52818229740471229188383712330, 14.42714853139638366484967255271, 15.42717251615517698216222628912, 15.932650905447351384820316491778, 16.844937822201368698477078967056, 17.67465274463075721007923377837, 18.28682437457015493800732941815, 19.36273270543394354555244733091, 20.51758547117214904282455921439, 21.067861174625963325794587313443, 21.659689455448511796732994315953, 22.64247011270029057029889486897