L(s) = 1 | − 5-s − 2.40·7-s + 5.38·11-s − 13-s + 7.17·17-s − 7.79·19-s − 2.40·23-s + 25-s − 4.97·29-s + 6.76·31-s + 2.40·35-s + 7.38·37-s − 3.38·41-s + 11.5·43-s − 10.7·47-s − 1.19·49-s − 1.43·53-s − 5.38·55-s + 5.94·59-s − 1.43·61-s + 65-s − 7.58·67-s − 2.61·71-s + 7.02·73-s − 12.9·77-s − 2.61·79-s + 1.94·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.910·7-s + 1.62·11-s − 0.277·13-s + 1.73·17-s − 1.78·19-s − 0.502·23-s + 0.200·25-s − 0.923·29-s + 1.21·31-s + 0.407·35-s + 1.21·37-s − 0.528·41-s + 1.76·43-s − 1.56·47-s − 0.171·49-s − 0.197·53-s − 0.725·55-s + 0.774·59-s − 0.183·61-s + 0.124·65-s − 0.926·67-s − 0.310·71-s + 0.822·73-s − 1.47·77-s − 0.294·79-s + 0.213·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.617898637\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.617898637\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + 2.40T + 7T^{2} \) |
| 11 | \( 1 - 5.38T + 11T^{2} \) |
| 17 | \( 1 - 7.17T + 17T^{2} \) |
| 19 | \( 1 + 7.79T + 19T^{2} \) |
| 23 | \( 1 + 2.40T + 23T^{2} \) |
| 29 | \( 1 + 4.97T + 29T^{2} \) |
| 31 | \( 1 - 6.76T + 31T^{2} \) |
| 37 | \( 1 - 7.38T + 37T^{2} \) |
| 41 | \( 1 + 3.38T + 41T^{2} \) |
| 43 | \( 1 - 11.5T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 + 1.43T + 53T^{2} \) |
| 59 | \( 1 - 5.94T + 59T^{2} \) |
| 61 | \( 1 + 1.43T + 61T^{2} \) |
| 67 | \( 1 + 7.58T + 67T^{2} \) |
| 71 | \( 1 + 2.61T + 71T^{2} \) |
| 73 | \( 1 - 7.02T + 73T^{2} \) |
| 79 | \( 1 + 2.61T + 79T^{2} \) |
| 83 | \( 1 - 1.94T + 83T^{2} \) |
| 89 | \( 1 + 0.618T + 89T^{2} \) |
| 97 | \( 1 - 9.22T + 97T^{2} \) |
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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
−7.72727758785496048815386891581, −6.95742686474323348865470709992, −6.22728022545938446297648614546, −6.00596817584269508096284005632, −4.79693731006936787764420299243, −4.04956922025407597211047153042, −3.58250902406739965140251172183, −2.74108496715956647860155590263, −1.65113065445710136077801380211, −0.62211252963728692915478558170,
0.62211252963728692915478558170, 1.65113065445710136077801380211, 2.74108496715956647860155590263, 3.58250902406739965140251172183, 4.04956922025407597211047153042, 4.79693731006936787764420299243, 6.00596817584269508096284005632, 6.22728022545938446297648614546, 6.95742686474323348865470709992, 7.72727758785496048815386891581