L(s) = 1 | + (0.654 − 0.755i)2-s + (−0.841 + 0.540i)3-s + (−0.142 − 0.989i)4-s + (−0.142 + 0.989i)6-s + (−0.959 + 0.281i)7-s + (−0.841 − 0.540i)8-s + (0.415 − 0.909i)9-s + (0.654 + 0.755i)11-s + (0.654 + 0.755i)12-s + (0.959 + 0.281i)13-s + (−0.415 + 0.909i)14-s + (−0.959 + 0.281i)16-s + (−0.142 + 0.989i)17-s + (−0.415 − 0.909i)18-s + (0.142 + 0.989i)19-s + ⋯ |
L(s) = 1 | + (0.654 − 0.755i)2-s + (−0.841 + 0.540i)3-s + (−0.142 − 0.989i)4-s + (−0.142 + 0.989i)6-s + (−0.959 + 0.281i)7-s + (−0.841 − 0.540i)8-s + (0.415 − 0.909i)9-s + (0.654 + 0.755i)11-s + (0.654 + 0.755i)12-s + (0.959 + 0.281i)13-s + (−0.415 + 0.909i)14-s + (−0.959 + 0.281i)16-s + (−0.142 + 0.989i)17-s + (−0.415 − 0.909i)18-s + (0.142 + 0.989i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.746 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.746 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.224975927 + 0.4661829782i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.224975927 + 0.4661829782i\) |
\(L(1)\) |
\(\approx\) |
\(1.023980081 - 0.08598847539i\) |
\(L(1)\) |
\(\approx\) |
\(1.023980081 - 0.08598847539i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + (0.654 - 0.755i)T \) |
| 3 | \( 1 + (-0.841 + 0.540i)T \) |
| 7 | \( 1 + (-0.959 + 0.281i)T \) |
| 11 | \( 1 + (0.654 + 0.755i)T \) |
| 13 | \( 1 + (0.959 + 0.281i)T \) |
| 17 | \( 1 + (-0.142 + 0.989i)T \) |
| 19 | \( 1 + (0.142 + 0.989i)T \) |
| 29 | \( 1 + (-0.142 + 0.989i)T \) |
| 31 | \( 1 + (0.841 + 0.540i)T \) |
| 37 | \( 1 + (0.415 - 0.909i)T \) |
| 41 | \( 1 + (0.415 + 0.909i)T \) |
| 43 | \( 1 + (0.841 - 0.540i)T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + (-0.959 + 0.281i)T \) |
| 59 | \( 1 + (-0.959 - 0.281i)T \) |
| 61 | \( 1 + (-0.841 - 0.540i)T \) |
| 67 | \( 1 + (-0.654 + 0.755i)T \) |
| 71 | \( 1 + (-0.654 + 0.755i)T \) |
| 73 | \( 1 + (0.142 + 0.989i)T \) |
| 79 | \( 1 + (0.959 + 0.281i)T \) |
| 83 | \( 1 + (0.415 - 0.909i)T \) |
| 89 | \( 1 + (-0.841 + 0.540i)T \) |
| 97 | \( 1 + (0.415 + 0.909i)T \) |
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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
−29.20239327549828170078511876187, −27.9245793835919763227235592040, −26.71310747539867458087544763164, −25.608773950015336621300872357942, −24.65278879726830754653870976227, −23.80956111081338380572483652610, −22.67518792855589760058654092469, −22.38585555737529541686071017052, −21.00759820352889452122866230913, −19.471795750808345612222467995760, −18.290688376726286200079542104050, −17.208514469748353136101157193095, −16.32134487132892435190603201111, −15.5663834814475395398675891955, −13.71085259356816104484939926534, −13.32113401966263137309963260529, −12.018470476409130967468468545435, −11.07088441033112363880683853184, −9.265460336891076125312564828182, −7.77377867186651414594674755309, −6.56181773932557382883080030107, −5.980320287848634049572629417620, −4.50496205290606548646168465840, −3.03145835158436181933526244046, −0.53415434808938891919218619968,
1.4153913971700594884414844509, 3.42400368573424751273593322957, 4.35473671512418420279037088436, 5.84494516318612254229191601202, 6.56044706201252420423486111610, 9.09277214008995646962785458213, 10.033145516217154069203886903113, 11.008298156461673881436618709689, 12.199006415888191910902814544167, 12.83930413731739387060737580721, 14.390534084008874304271704183228, 15.49101601497922806098956910335, 16.43176871665979883719796999030, 17.84375845860935777781888112667, 18.942578593198225424819444217160, 20.06330907118872157094421897725, 21.169123998537553193151584967101, 22.03356724985623646818924029886, 22.89805074193550188714083385509, 23.48145962792323621122398207434, 24.93269320091232641285485374353, 26.25109133002141384401510959545, 27.62465506971972749823535073592, 28.317042810083933528036002865433, 29.03009638279592523995142271602