L(s) = 1 | + (−0.654 − 0.755i)3-s + (0.959 + 0.281i)5-s + (0.959 + 0.281i)7-s + (−0.142 + 0.989i)9-s + (−0.959 + 0.281i)11-s + (0.654 + 0.755i)13-s + (−0.415 − 0.909i)15-s + (0.415 − 0.909i)17-s + (−0.142 + 0.989i)19-s + (−0.415 − 0.909i)21-s + (0.142 − 0.989i)23-s + (0.841 + 0.540i)25-s + (0.841 − 0.540i)27-s + (0.959 + 0.281i)29-s + (0.142 + 0.989i)31-s + ⋯ |
L(s) = 1 | + (−0.654 − 0.755i)3-s + (0.959 + 0.281i)5-s + (0.959 + 0.281i)7-s + (−0.142 + 0.989i)9-s + (−0.959 + 0.281i)11-s + (0.654 + 0.755i)13-s + (−0.415 − 0.909i)15-s + (0.415 − 0.909i)17-s + (−0.142 + 0.989i)19-s + (−0.415 − 0.909i)21-s + (0.142 − 0.989i)23-s + (0.841 + 0.540i)25-s + (0.841 − 0.540i)27-s + (0.959 + 0.281i)29-s + (0.142 + 0.989i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.705782718 + 0.9528310993i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.705782718 + 0.9528310993i\) |
\(L(1)\) |
\(\approx\) |
\(1.122464496 + 0.04632429883i\) |
\(L(1)\) |
\(\approx\) |
\(1.122464496 + 0.04632429883i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 89 | \( 1 \) |
good | 3 | \( 1 + (-0.654 - 0.755i)T \) |
| 5 | \( 1 + (0.959 + 0.281i)T \) |
| 7 | \( 1 + (0.959 + 0.281i)T \) |
| 11 | \( 1 + (-0.959 + 0.281i)T \) |
| 13 | \( 1 + (0.654 + 0.755i)T \) |
| 17 | \( 1 + (0.415 - 0.909i)T \) |
| 19 | \( 1 + (-0.142 + 0.989i)T \) |
| 23 | \( 1 + (0.142 - 0.989i)T \) |
| 29 | \( 1 + (0.959 + 0.281i)T \) |
| 31 | \( 1 + (0.142 + 0.989i)T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + (-0.654 + 0.755i)T \) |
| 43 | \( 1 + (-0.959 + 0.281i)T \) |
| 47 | \( 1 + (0.654 - 0.755i)T \) |
| 53 | \( 1 + (0.654 + 0.755i)T \) |
| 59 | \( 1 + (-0.654 + 0.755i)T \) |
| 61 | \( 1 + (-0.841 + 0.540i)T \) |
| 67 | \( 1 + (-0.654 - 0.755i)T \) |
| 71 | \( 1 + (0.959 - 0.281i)T \) |
| 73 | \( 1 + (-0.142 - 0.989i)T \) |
| 79 | \( 1 + (0.142 + 0.989i)T \) |
| 83 | \( 1 + (0.415 - 0.909i)T \) |
| 97 | \( 1 + (-0.959 - 0.281i)T \) |
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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
−21.98766534515971580396244255190, −21.43752914056301081288857952576, −20.852735089939611488727462879623, −20.2368491110551331942731851092, −18.81657231696713474322894474551, −17.73123711390026179468817269051, −17.53215669845006645224238701072, −16.70525173406793116478262388342, −15.60216384695588222034829673862, −15.12414027390071213789247837932, −13.90205892684888899020457946111, −13.26649968057438554326706853097, −12.23872566125495054380480500704, −11.12220271225186520657727537547, −10.57571001988739883964427964588, −9.89186861525082756298458242679, −8.768141691170483506179088502163, −8.02213068856954907670003734383, −6.64730902412175689395915758488, −5.5373962854132210389512598817, −5.25401795195542146610840759653, −4.14770191142424768249350005130, −2.96060740298831301741495908538, −1.59673488295203595737920971815, −0.51303545371340131864609331799,
1.18371842115600220733997377702, 1.932839934578364202985157322546, 2.88072816023209604149617661008, 4.7339857044831415250665760537, 5.31168350812619945694933748927, 6.25739008773874602963733490849, 7.04490420448016872445735111524, 8.05691672419055103072427830597, 8.886260631343311117009642895612, 10.31548823064526861347932107197, 10.69681485690642646135561367380, 11.86280416060337265891481402131, 12.41452212260675529613613197708, 13.663772387703649625070019307221, 13.94331082144045856491128892077, 15.00188723173561819522570205325, 16.26103576247044347587226094801, 16.88423787655984474931442991688, 17.94575262187152242627243738903, 18.32429120616164018184182207735, 18.78209153811312091212242267288, 20.23007476471380923475778196609, 21.18812006660069434661292028416, 21.49458353813184734446262924384, 22.75314294811476063761365425487