L(s) = 1 | + (0.600 + 0.600i)3-s + (1.01 + 1.01i)7-s − 2.27i·9-s − 2.07i·11-s + (−0.221 − 0.221i)13-s + (−1.58 − 1.58i)17-s + 8.07·19-s + 1.21i·21-s + (−4.78 − 0.356i)23-s + (3.17 − 3.17i)27-s − 3.88i·29-s − 3.40·31-s + (1.24 − 1.24i)33-s + (1.10 + 1.10i)37-s − 0.265i·39-s + ⋯ |
L(s) = 1 | + (0.346 + 0.346i)3-s + (0.382 + 0.382i)7-s − 0.759i·9-s − 0.626i·11-s + (−0.0613 − 0.0613i)13-s + (−0.383 − 0.383i)17-s + 1.85·19-s + 0.265i·21-s + (−0.997 − 0.0743i)23-s + (0.610 − 0.610i)27-s − 0.721i·29-s − 0.612·31-s + (0.217 − 0.217i)33-s + (0.182 + 0.182i)37-s − 0.0425i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 + 0.665i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.989121978\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.989121978\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + (4.78 + 0.356i)T \) |
good | 3 | \( 1 + (-0.600 - 0.600i)T + 3iT^{2} \) |
| 7 | \( 1 + (-1.01 - 1.01i)T + 7iT^{2} \) |
| 11 | \( 1 + 2.07iT - 11T^{2} \) |
| 13 | \( 1 + (0.221 + 0.221i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.58 + 1.58i)T + 17iT^{2} \) |
| 19 | \( 1 - 8.07T + 19T^{2} \) |
| 29 | \( 1 + 3.88iT - 29T^{2} \) |
| 31 | \( 1 + 3.40T + 31T^{2} \) |
| 37 | \( 1 + (-1.10 - 1.10i)T + 37iT^{2} \) |
| 41 | \( 1 + 2.60T + 41T^{2} \) |
| 43 | \( 1 + (-4.22 + 4.22i)T - 43iT^{2} \) |
| 47 | \( 1 + (-3.57 + 3.57i)T - 47iT^{2} \) |
| 53 | \( 1 + (3.65 - 3.65i)T - 53iT^{2} \) |
| 59 | \( 1 + 8.97iT - 59T^{2} \) |
| 61 | \( 1 + 8.43iT - 61T^{2} \) |
| 67 | \( 1 + (-7.20 - 7.20i)T + 67iT^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 + (-5.96 - 5.96i)T + 73iT^{2} \) |
| 79 | \( 1 - 5.02T + 79T^{2} \) |
| 83 | \( 1 + (-8.93 + 8.93i)T - 83iT^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 + (1.91 + 1.91i)T + 97iT^{2} \) |
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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
−9.008632557247611926107660906022, −8.219857717082904973656493267373, −7.51011149732758892319914796663, −6.54421942086170804084829326418, −5.71529039278126006902036634110, −4.98737808742114437854347685590, −3.87963911419863624041728483771, −3.22634967428508272218205940591, −2.15465287026242714638376710017, −0.70117200797015327331874040913,
1.29322948069448412650746566230, 2.18661484124322368418029047891, 3.25469204175709074664176207311, 4.32400946830837132015811016155, 5.07779694445161716284791530120, 5.93297758403395069005932884071, 7.08331377106989196581847693362, 7.56468997607753031971633632607, 8.127588533998762902962369689785, 9.104233581742972087051977132603