L(s) = 1 | + (−0.939 + 0.342i)3-s + (−0.173 + 0.984i)5-s + (0.5 + 0.866i)7-s + (0.766 − 0.642i)9-s + (−0.5 + 0.866i)11-s + (0.939 + 0.342i)13-s + (−0.173 − 0.984i)15-s + (0.766 + 0.642i)17-s + (−0.766 − 0.642i)21-s + (−0.173 − 0.984i)23-s + (−0.939 − 0.342i)25-s + (−0.5 + 0.866i)27-s + (−0.766 + 0.642i)29-s + (0.5 + 0.866i)31-s + (0.173 − 0.984i)33-s + ⋯ |
L(s) = 1 | + (−0.939 + 0.342i)3-s + (−0.173 + 0.984i)5-s + (0.5 + 0.866i)7-s + (0.766 − 0.642i)9-s + (−0.5 + 0.866i)11-s + (0.939 + 0.342i)13-s + (−0.173 − 0.984i)15-s + (0.766 + 0.642i)17-s + (−0.766 − 0.642i)21-s + (−0.173 − 0.984i)23-s + (−0.939 − 0.342i)25-s + (−0.5 + 0.866i)27-s + (−0.766 + 0.642i)29-s + (0.5 + 0.866i)31-s + (0.173 − 0.984i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1016404541 + 0.9484893750i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1016404541 + 0.9484893750i\) |
\(L(1)\) |
\(\approx\) |
\(0.6721367430 + 0.4222963984i\) |
\(L(1)\) |
\(\approx\) |
\(0.6721367430 + 0.4222963984i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (-0.939 + 0.342i)T \) |
| 5 | \( 1 + (-0.173 + 0.984i)T \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (0.939 + 0.342i)T \) |
| 17 | \( 1 + (0.766 + 0.642i)T \) |
| 23 | \( 1 + (-0.173 - 0.984i)T \) |
| 29 | \( 1 + (-0.766 + 0.642i)T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + (-0.939 + 0.342i)T \) |
| 43 | \( 1 + (0.173 - 0.984i)T \) |
| 47 | \( 1 + (-0.766 + 0.642i)T \) |
| 53 | \( 1 + (-0.173 - 0.984i)T \) |
| 59 | \( 1 + (0.766 + 0.642i)T \) |
| 61 | \( 1 + (-0.173 - 0.984i)T \) |
| 67 | \( 1 + (0.766 - 0.642i)T \) |
| 71 | \( 1 + (-0.173 + 0.984i)T \) |
| 73 | \( 1 + (-0.939 + 0.342i)T \) |
| 79 | \( 1 + (0.939 - 0.342i)T \) |
| 83 | \( 1 + (-0.5 - 0.866i)T \) |
| 89 | \( 1 + (-0.939 - 0.342i)T \) |
| 97 | \( 1 + (0.766 + 0.642i)T \) |
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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
−27.67159536475679689938623755203, −26.56038840695296703113689939350, −25.119193568815543828237722307520, −24.11370573780905634926384296912, −23.57985789704304021119536503975, −22.708205426765714125253169171843, −21.2231219690442378825014954110, −20.64759647525656760895917632335, −19.29922448533066127193344040416, −18.2370642106606979908076403034, −17.19843277200271061847465477971, −16.472921097949737923337756584950, −15.61892221135750168703369641593, −13.69610575193750912087492484590, −13.18931319123129733476508001495, −11.82621980093412424103746256045, −11.0985229252006948440301921702, −9.91256648381889272250982075979, −8.28891022480181636814500362955, −7.474945850524891823771159566465, −5.89308997298828081297786646408, −5.05379187491369278297404149367, −3.7595115039368366119613080215, −1.40964368304635516115277071917, −0.439863178766526273055229799104,
1.81089393402381411105565986576, 3.498077155031194324369483300582, 4.90268531688664449549481582295, 6.01969004389261658338456155930, 7.0358683812406125165836719895, 8.46143483384154094151396407943, 10.00071863076335073845441181426, 10.809318276523312199735086271730, 11.777763570000237485176375019722, 12.68499434576016197320646418849, 14.395114157360810172973316596609, 15.25404802369424555168372521957, 16.079726937683383509415242109872, 17.44743065959413895048439232279, 18.3024074021478390302405613686, 18.87920313131559389997135222511, 20.67863955924737163856079474776, 21.48923508536379065594118605397, 22.4207756787263045746647148962, 23.210907651369464106279819278339, 24.06242906476539201961464954465, 25.51530144500325230750057843672, 26.316101769281603322004807190266, 27.50034668698642548415942450841, 28.17157917178677685285110148670