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
−28.17157917178677685285110148670, −27.50034668698642548415942450841, −26.316101769281603322004807190266, −25.51530144500325230750057843672, −24.06242906476539201961464954465, −23.210907651369464106279819278339, −22.4207756787263045746647148962, −21.48923508536379065594118605397, −20.67863955924737163856079474776, −18.87920313131559389997135222511, −18.3024074021478390302405613686, −17.44743065959413895048439232279, −16.079726937683383509415242109872, −15.25404802369424555168372521957, −14.395114157360810172973316596609, −12.68499434576016197320646418849, −11.777763570000237485176375019722, −10.809318276523312199735086271730, −10.00071863076335073845441181426, −8.46143483384154094151396407943, −7.0358683812406125165836719895, −6.01969004389261658338456155930, −4.90268531688664449549481582295, −3.498077155031194324369483300582, −1.81089393402381411105565986576,
0.439863178766526273055229799104, 1.40964368304635516115277071917, 3.7595115039368366119613080215, 5.05379187491369278297404149367, 5.89308997298828081297786646408, 7.474945850524891823771159566465, 8.28891022480181636814500362955, 9.91256648381889272250982075979, 11.0985229252006948440301921702, 11.82621980093412424103746256045, 13.18931319123129733476508001495, 13.69610575193750912087492484590, 15.61892221135750168703369641593, 16.472921097949737923337756584950, 17.19843277200271061847465477971, 18.2370642106606979908076403034, 19.29922448533066127193344040416, 20.64759647525656760895917632335, 21.2231219690442378825014954110, 22.708205426765714125253169171843, 23.57985789704304021119536503975, 24.11370573780905634926384296912, 25.119193568815543828237722307520, 26.56038840695296703113689939350, 27.67159536475679689938623755203