L(s) = 1 | + (0.748 − 0.663i)2-s + (−0.354 − 0.935i)3-s + (0.120 − 0.992i)4-s + (0.970 − 0.239i)5-s + (−0.885 − 0.464i)6-s + (−0.568 − 0.822i)7-s + (−0.568 − 0.822i)8-s + (−0.748 + 0.663i)9-s + (0.568 − 0.822i)10-s + (0.748 + 0.663i)11-s + (−0.970 + 0.239i)12-s + (−0.970 − 0.239i)14-s + (−0.568 − 0.822i)15-s + (−0.970 − 0.239i)16-s + (0.568 + 0.822i)17-s + (−0.120 + 0.992i)18-s + ⋯ |
L(s) = 1 | + (0.748 − 0.663i)2-s + (−0.354 − 0.935i)3-s + (0.120 − 0.992i)4-s + (0.970 − 0.239i)5-s + (−0.885 − 0.464i)6-s + (−0.568 − 0.822i)7-s + (−0.568 − 0.822i)8-s + (−0.748 + 0.663i)9-s + (0.568 − 0.822i)10-s + (0.748 + 0.663i)11-s + (−0.970 + 0.239i)12-s + (−0.970 − 0.239i)14-s + (−0.568 − 0.822i)15-s + (−0.970 − 0.239i)16-s + (0.568 + 0.822i)17-s + (−0.120 + 0.992i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.760 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.760 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5488669101 - 1.488834150i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5488669101 - 1.488834150i\) |
\(L(1)\) |
\(\approx\) |
\(0.9995782133 - 1.056271103i\) |
\(L(1)\) |
\(\approx\) |
\(0.9995782133 - 1.056271103i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
good | 2 | \( 1 + (0.748 - 0.663i)T \) |
| 3 | \( 1 + (-0.354 - 0.935i)T \) |
| 5 | \( 1 + (0.970 - 0.239i)T \) |
| 7 | \( 1 + (-0.568 - 0.822i)T \) |
| 11 | \( 1 + (0.748 + 0.663i)T \) |
| 17 | \( 1 + (0.568 + 0.822i)T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (-0.748 + 0.663i)T \) |
| 31 | \( 1 + (-0.885 - 0.464i)T \) |
| 37 | \( 1 + (-0.885 - 0.464i)T \) |
| 41 | \( 1 + (0.354 + 0.935i)T \) |
| 43 | \( 1 + (0.885 - 0.464i)T \) |
| 47 | \( 1 + (-0.120 - 0.992i)T \) |
| 53 | \( 1 + (0.568 + 0.822i)T \) |
| 59 | \( 1 + (0.970 - 0.239i)T \) |
| 61 | \( 1 + (0.568 - 0.822i)T \) |
| 67 | \( 1 + (-0.120 - 0.992i)T \) |
| 71 | \( 1 + (0.354 + 0.935i)T \) |
| 73 | \( 1 + (0.748 + 0.663i)T \) |
| 79 | \( 1 + (0.120 + 0.992i)T \) |
| 83 | \( 1 + (0.354 - 0.935i)T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (0.970 + 0.239i)T \) |
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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
−27.80922241539787191747506677717, −26.85462866528447370018770089937, −25.70852195298982905549296572380, −25.29349916566218618212875971926, −24.141846557401412778096005066301, −22.63381349754312802161543971149, −22.40248241709906693956680833826, −21.32413972298356473275923087284, −20.88628022929692447101194849652, −19.09160908137793667575560333534, −17.73738256479061376080605690338, −16.84095801112074209169198281704, −16.12504675771875820470448282085, −14.98676762160294788267396026686, −14.30356977614205494320657808247, −13.10948307410345644591755911224, −11.97607291348991476015265561710, −10.88209791835113341140354286047, −9.420652024701885522453846777154, −8.77903939562383372871485190127, −6.79449806527534866350637946718, −5.89315880484074041959399190215, −5.19666560513188131514864935361, −3.69654526958600501097600644657, −2.64125294165239164643381932276,
1.1946439562728630218396939068, 2.19572999824867675070525805765, 3.797906838103397959140852238686, 5.25112651347787829977524506150, 6.303894894239993455342501689091, 7.08981560347012374421431060667, 9.04302932027460773683961119197, 10.21313875614632377591957016151, 11.10968408319318904283664435776, 12.66354857143613075130083864483, 12.834945189275112231638885559382, 13.97257650952746435011543330127, 14.78632458939455931083548481770, 16.69214496416962086314745825263, 17.33186683608120621847941534067, 18.607724082278583310097975879050, 19.53626519617566043620239283980, 20.34193612397239336841694236809, 21.48336198443425000482625523647, 22.50018749642745248225729786315, 23.23095541473116944232374722605, 24.1113537094448165691400578266, 25.128034036546048762789256334337, 25.871365821738932344873918964727, 27.7367811077945556724185575161