L(s) = 1 | + (0.939 + 0.342i)2-s + (0.173 − 0.984i)3-s + (0.766 + 0.642i)4-s + (0.5 − 0.866i)6-s + (0.500 + 0.866i)8-s + (−0.939 − 0.342i)9-s + (−1.70 + 0.984i)11-s + (0.766 − 0.642i)12-s + (0.173 + 0.984i)16-s + (0.592 − 1.62i)17-s + (−0.766 − 0.642i)18-s + (−0.766 − 0.642i)19-s + (−1.93 + 0.342i)22-s + (0.939 − 0.342i)24-s + (−0.173 + 0.984i)25-s + ⋯ |
L(s) = 1 | + (0.939 + 0.342i)2-s + (0.173 − 0.984i)3-s + (0.766 + 0.642i)4-s + (0.5 − 0.866i)6-s + (0.500 + 0.866i)8-s + (−0.939 − 0.342i)9-s + (−1.70 + 0.984i)11-s + (0.766 − 0.642i)12-s + (0.173 + 0.984i)16-s + (0.592 − 1.62i)17-s + (−0.766 − 0.642i)18-s + (−0.766 − 0.642i)19-s + (−1.93 + 0.342i)22-s + (0.939 − 0.342i)24-s + (−0.173 + 0.984i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.400501036\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.400501036\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.939 - 0.342i)T \) |
| 3 | \( 1 + (-0.173 + 0.984i)T \) |
| 19 | \( 1 + (0.766 + 0.642i)T \) |
good | 5 | \( 1 + (0.173 - 0.984i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (1.70 - 0.984i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 17 | \( 1 + (-0.592 + 1.62i)T + (-0.766 - 0.642i)T^{2} \) |
| 23 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 29 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \) |
| 43 | \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 47 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 53 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 59 | \( 1 + (-0.326 - 0.118i)T + (0.766 + 0.642i)T^{2} \) |
| 61 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 67 | \( 1 + (0.439 + 1.20i)T + (-0.766 + 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 73 | \( 1 + (0.0603 + 0.342i)T + (-0.939 + 0.342i)T^{2} \) |
| 79 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 83 | \( 1 + (-1.11 - 0.642i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \) |
| 97 | \( 1 + (0.439 - 1.20i)T + (-0.766 - 0.642i)T^{2} \) |
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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
−11.58644033804605129304195688490, −10.67531827338755564451744493589, −9.371442260815242085057194653343, −8.042032750089009665627504528303, −7.47291632305022087515202235906, −6.75095672375867351603708165988, −5.50734219663028214667740218416, −4.79357235310370281409538590780, −3.07323738108844031390228529974, −2.22450802316928221900770167198,
2.40413955278088361217354288840, 3.48047389258595842018182408685, 4.40013635516332379903188780361, 5.58188382645740824915253304497, 6.05507416116499818873264579333, 7.81801130376256686063151760639, 8.541839343564402169167972820196, 9.995462462372261982147277781364, 10.57327597221300734412038679539, 11.02663714980886234298396075767