| L(s) = 1 | + 3-s + (0.891 + 0.453i)5-s + (0.156 − 0.987i)7-s + 9-s + (−0.951 − 0.309i)11-s + (−0.809 − 0.587i)13-s + (0.891 + 0.453i)15-s + (−0.453 − 0.891i)17-s + (−0.587 − 0.809i)19-s + (0.156 − 0.987i)21-s + (−0.587 + 0.809i)23-s + (0.587 + 0.809i)25-s + 27-s + (−0.309 − 0.951i)29-s + (0.309 − 0.951i)31-s + ⋯ |
| L(s) = 1 | + 3-s + (0.891 + 0.453i)5-s + (0.156 − 0.987i)7-s + 9-s + (−0.951 − 0.309i)11-s + (−0.809 − 0.587i)13-s + (0.891 + 0.453i)15-s + (−0.453 − 0.891i)17-s + (−0.587 − 0.809i)19-s + (0.156 − 0.987i)21-s + (−0.587 + 0.809i)23-s + (0.587 + 0.809i)25-s + 27-s + (−0.309 − 0.951i)29-s + (0.309 − 0.951i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1312 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.361 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1312 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.361 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.886474873 - 1.291676128i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.886474873 - 1.291676128i\) |
| \(L(1)\) |
\(\approx\) |
\(1.527844760 - 0.3333951322i\) |
| \(L(1)\) |
\(\approx\) |
\(1.527844760 - 0.3333951322i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 41 | \( 1 \) |
| good | 3 | \( 1 + T \) |
| 5 | \( 1 + (0.891 + 0.453i)T \) |
| 7 | \( 1 + (0.156 - 0.987i)T \) |
| 11 | \( 1 + (-0.951 - 0.309i)T \) |
| 13 | \( 1 + (-0.809 - 0.587i)T \) |
| 17 | \( 1 + (-0.453 - 0.891i)T \) |
| 19 | \( 1 + (-0.587 - 0.809i)T \) |
| 23 | \( 1 + (-0.587 + 0.809i)T \) |
| 29 | \( 1 + (-0.309 - 0.951i)T \) |
| 31 | \( 1 + (0.309 - 0.951i)T \) |
| 37 | \( 1 + (0.453 - 0.891i)T \) |
| 43 | \( 1 + (0.987 - 0.156i)T \) |
| 47 | \( 1 + (0.987 - 0.156i)T \) |
| 53 | \( 1 + (-0.951 + 0.309i)T \) |
| 59 | \( 1 + (-0.156 - 0.987i)T \) |
| 61 | \( 1 + (0.987 + 0.156i)T \) |
| 67 | \( 1 + (0.951 - 0.309i)T \) |
| 71 | \( 1 + (0.453 + 0.891i)T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + (0.707 + 0.707i)T \) |
| 83 | \( 1 + (0.707 - 0.707i)T \) |
| 89 | \( 1 + (-0.156 + 0.987i)T \) |
| 97 | \( 1 + (0.891 + 0.453i)T \) |
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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
−20.996345667090439092333126032418, −20.54692638217607254249810308550, −19.58697258526520986155790252160, −18.811272948641442325875243299084, −18.20454730233085541171164476000, −17.42130095219797103065505554035, −16.41813295579672452102398532658, −15.692228932197565776454636667997, −14.75292927683139258304840049081, −14.38308322253738148217408628946, −13.40147135166566202457326673142, −12.53280980227906057092301812073, −12.34663990950569207319122876757, −10.70674608850217559419652599879, −10.04563187344839609404664463023, −9.27031457855565514276063281442, −8.55999104556435455515880643233, −8.001822393703819445829646149201, −6.82076277781056383997941114852, −5.945150427732765256496401654785, −4.96866766617147874650354780949, −4.26861581326580225362713298095, −2.87484658468462179902694910475, −2.19051379217544375037466381640, −1.62001305155800710938727119345,
0.716232170361144588862355146716, 2.2711931453758283720808545151, 2.5310545036638508764630441037, 3.68561718584787213723779400906, 4.63530139961648051121791310251, 5.572243024892648052842192002716, 6.71981827474278597978178931601, 7.51229608260888147654775354616, 7.995033669529650433624299629730, 9.2873668353359166502063294395, 9.76738263988252841173456853786, 10.555289413526637737172476443380, 11.23674143824928324752617000688, 12.70484238602154926221159485837, 13.39147484148199957943164757087, 13.77824401766272487505851483041, 14.55492655008970451865150472680, 15.36803076493941925522123087852, 16.05630575324735009339202219288, 17.32080377010173102106362005403, 17.65060531068783472489861933189, 18.65031943890000727490136712417, 19.29910141251152434039989572137, 20.25543056961230318472285326456, 20.64136401717215714300416612291
