L(s) = 1 | + 2.21i·3-s + 5-s + (−1.20 − 2.35i)7-s − 1.92·9-s + 3.88·11-s − 5.67·13-s + 2.21i·15-s + 5.63i·17-s + 1.31i·19-s + (5.22 − 2.68i)21-s + 7.37i·23-s + 25-s + 2.38i·27-s + 9.07i·29-s + 2.23·31-s + ⋯ |
L(s) = 1 | + 1.28i·3-s + 0.447·5-s + (−0.457 − 0.889i)7-s − 0.641·9-s + 1.17·11-s − 1.57·13-s + 0.572i·15-s + 1.36i·17-s + 0.300i·19-s + (1.13 − 0.585i)21-s + 1.53i·23-s + 0.200·25-s + 0.459i·27-s + 1.68i·29-s + 0.400·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.504 - 0.863i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.504 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.452417791\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.452417791\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (1.20 + 2.35i)T \) |
good | 3 | \( 1 - 2.21iT - 3T^{2} \) |
| 11 | \( 1 - 3.88T + 11T^{2} \) |
| 13 | \( 1 + 5.67T + 13T^{2} \) |
| 17 | \( 1 - 5.63iT - 17T^{2} \) |
| 19 | \( 1 - 1.31iT - 19T^{2} \) |
| 23 | \( 1 - 7.37iT - 23T^{2} \) |
| 29 | \( 1 - 9.07iT - 29T^{2} \) |
| 31 | \( 1 - 2.23T + 31T^{2} \) |
| 37 | \( 1 + 6.98iT - 37T^{2} \) |
| 41 | \( 1 - 7.47iT - 41T^{2} \) |
| 43 | \( 1 - 1.46T + 43T^{2} \) |
| 47 | \( 1 + 0.567T + 47T^{2} \) |
| 53 | \( 1 + 0.100iT - 53T^{2} \) |
| 59 | \( 1 + 2.93iT - 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 + 5.54T + 67T^{2} \) |
| 71 | \( 1 - 2.42iT - 71T^{2} \) |
| 73 | \( 1 + 6.08iT - 73T^{2} \) |
| 79 | \( 1 + 2.83iT - 79T^{2} \) |
| 83 | \( 1 - 2.52iT - 83T^{2} \) |
| 89 | \( 1 + 10.9iT - 89T^{2} \) |
| 97 | \( 1 + 9.93iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.957724322205110308007666758312, −9.575562115019866585116788846836, −8.808170379612836949021588763881, −7.53219927950420473177794288905, −6.78060947961213427456850076986, −5.73907869234775131054124996537, −4.78379687035194463605901582788, −3.96091181001566835037586608868, −3.27683141211025921504582063759, −1.57539777070236663583569894144,
0.64496432463255853695470338133, 2.22260199789315473179877439408, 2.68456960736280826968484437464, 4.43260170479840404164654942735, 5.45735874322511628775584529836, 6.54218232507875070350261199393, 6.80304992420518118456440892448, 7.78851411323115534116817775293, 8.752206709935286666497747616048, 9.521542628702478206102157598240