L(s) = 1 | + 2-s + (0.173 − 0.984i)3-s + 4-s + (0.939 − 0.342i)5-s + (0.173 − 0.984i)6-s + (−0.939 + 0.342i)7-s + 8-s + (−0.939 − 0.342i)9-s + (0.939 − 0.342i)10-s + (0.939 + 0.342i)11-s + (0.173 − 0.984i)12-s + (−0.939 + 0.342i)13-s + (−0.939 + 0.342i)14-s + (−0.173 − 0.984i)15-s + 16-s + 17-s + ⋯ |
L(s) = 1 | + 2-s + (0.173 − 0.984i)3-s + 4-s + (0.939 − 0.342i)5-s + (0.173 − 0.984i)6-s + (−0.939 + 0.342i)7-s + 8-s + (−0.939 − 0.342i)9-s + (0.939 − 0.342i)10-s + (0.939 + 0.342i)11-s + (0.173 − 0.984i)12-s + (−0.939 + 0.342i)13-s + (−0.939 + 0.342i)14-s + (−0.173 − 0.984i)15-s + 16-s + 17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0846 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0846 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.801235901 - 3.049292207i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.801235901 - 3.049292207i\) |
\(L(1)\) |
\(\approx\) |
\(2.068879410 - 0.9325037771i\) |
\(L(1)\) |
\(\approx\) |
\(2.068879410 - 0.9325037771i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + T \) |
| 3 | \( 1 + (0.173 - 0.984i)T \) |
| 5 | \( 1 + (0.939 - 0.342i)T \) |
| 7 | \( 1 + (-0.939 + 0.342i)T \) |
| 11 | \( 1 + (0.939 + 0.342i)T \) |
| 13 | \( 1 + (-0.939 + 0.342i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.939 - 0.342i)T \) |
| 31 | \( 1 + (-0.766 + 0.642i)T \) |
| 41 | \( 1 + (0.5 - 0.866i)T \) |
| 43 | \( 1 + (0.5 - 0.866i)T \) |
| 47 | \( 1 + (-0.173 - 0.984i)T \) |
| 53 | \( 1 + (-0.173 + 0.984i)T \) |
| 59 | \( 1 + (0.173 + 0.984i)T \) |
| 61 | \( 1 + (-0.766 - 0.642i)T \) |
| 67 | \( 1 + (-0.173 - 0.984i)T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + (-0.939 - 0.342i)T \) |
| 79 | \( 1 + (0.173 + 0.984i)T \) |
| 83 | \( 1 + (0.766 + 0.642i)T \) |
| 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
−19.039808168456291884490912295620, −17.6045793056300087834420777220, −17.06808412956628630477484319334, −16.351011962478510318852854607336, −16.09446712012426109621208100351, −14.876001992873282354438858706360, −14.57357032110183643200591723657, −14.08203448481252512999316965671, −13.31505876436948291646161779853, −12.589333203787728527224189649325, −11.87473012151712641258144628389, −11.013326294961568751392010181, −10.314204715392540846430107490184, −9.77044035928659841773018701602, −9.359628434301112830171968869235, −8.080018841446194731678670767834, −7.30169592174213506597388460303, −6.25280270483185896039232169690, −5.990858003601824351585415850011, −5.20330002455900926015797002697, −4.33779291709516629262259976078, −3.53995356799744911801887919654, −3.10408207693458745107267836554, −2.30161385632212991917012771568, −1.24660071154490690367873536900,
0.742796631306617053119129645721, 1.79580284723902033285733533048, 2.36646905613720416067954477796, 2.988648889576363784988859195269, 3.980701777597621736656162227447, 4.92615934033878702758935110386, 5.691541560369935396169911557750, 6.33191030963603758568850743820, 6.81979656563336702734461098314, 7.42913395547374721549247612222, 8.590653593684595922694527191205, 9.24472467355425622775438742152, 9.99933617007130933037672494061, 10.8040038533287362467875071014, 12.02649624470172177953378133487, 12.31963936089995764756300213232, 12.64837400353216031314833329235, 13.63203810475501186510165963651, 14.00358798641457995229654753808, 14.62201461089137019914924152756, 15.326209564018907236993260935391, 16.41769984537916400098813261576, 16.87131186586528763418957332247, 17.44172134131088826117117587169, 18.35581579053547634569823736146