L(s) = 1 | + (−0.587 + 0.809i)2-s + (1.69 − 0.550i)3-s + (−0.309 − 0.951i)4-s + (−0.987 + 0.156i)5-s + (−0.550 + 1.69i)6-s + i·7-s + (0.951 + 0.309i)8-s + (1.76 − 1.27i)9-s + (0.453 − 0.891i)10-s + (−1.04 − 1.44i)12-s + (0.533 + 0.734i)13-s + (−0.809 − 0.587i)14-s + (−1.58 + 0.809i)15-s + (−0.809 + 0.587i)16-s + 2.17i·18-s + (−0.437 + 1.34i)19-s + ⋯ |
L(s) = 1 | + (−0.587 + 0.809i)2-s + (1.69 − 0.550i)3-s + (−0.309 − 0.951i)4-s + (−0.987 + 0.156i)5-s + (−0.550 + 1.69i)6-s + i·7-s + (0.951 + 0.309i)8-s + (1.76 − 1.27i)9-s + (0.453 − 0.891i)10-s + (−1.04 − 1.44i)12-s + (0.533 + 0.734i)13-s + (−0.809 − 0.587i)14-s + (−1.58 + 0.809i)15-s + (−0.809 + 0.587i)16-s + 2.17i·18-s + (−0.437 + 1.34i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.233589120\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.233589120\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.587 - 0.809i)T \) |
| 5 | \( 1 + (0.987 - 0.156i)T \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 + (-1.69 + 0.550i)T + (0.809 - 0.587i)T^{2} \) |
| 11 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.533 - 0.734i)T + (-0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (0.437 - 1.34i)T + (-0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (-1.11 + 1.53i)T + (-0.309 - 0.951i)T^{2} \) |
| 29 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.253 + 0.183i)T + (0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (1.59 + 1.16i)T + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 71 | \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (-0.587 - 1.80i)T + (-0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (1.87 + 0.610i)T + (0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 97 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
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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
−9.332171923033711081333646713454, −8.746628206175118287976397845247, −8.329974540434201912020050406153, −7.71238128341922892681809801741, −6.83771887591147546938983384412, −6.20959641004649640617118023101, −4.72070903555967700015510676176, −3.77238448088961556706744701029, −2.67121815734096461547153839704, −1.58018785390690581375111556196,
1.25997963002199621386917701032, 2.81623161667888085764247650052, 3.39690627252476317691984265270, 4.11221564691051548823149651257, 4.84292855668831258438538444586, 7.13820792968378167863682420903, 7.50581553574832835141140305079, 8.280599055089509959678175842569, 8.894955648713404683617235246664, 9.481014439731990175536634711217