L(s) = 1 | − 3-s − 0.508·5-s − 3.87·7-s + 9-s − 11-s − 0.508·13-s + 0.508·15-s + 7.36·17-s − 5.36·19-s + 3.87·21-s − 2.50·23-s − 4.74·25-s − 27-s − 7.36·29-s + 10.7·31-s + 33-s + 1.96·35-s − 8.72·37-s + 0.508·39-s + 8.37·41-s + 2.37·43-s − 0.508·45-s − 5.49·47-s + 7.98·49-s − 7.36·51-s + 12.2·53-s + 0.508·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.227·5-s − 1.46·7-s + 0.333·9-s − 0.301·11-s − 0.140·13-s + 0.131·15-s + 1.78·17-s − 1.23·19-s + 0.844·21-s − 0.522·23-s − 0.948·25-s − 0.192·27-s − 1.36·29-s + 1.92·31-s + 0.174·33-s + 0.332·35-s − 1.43·37-s + 0.0813·39-s + 1.30·41-s + 0.362·43-s − 0.0757·45-s − 0.801·47-s + 1.14·49-s − 1.03·51-s + 1.68·53-s + 0.0685·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8358051199\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8358051199\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 + 0.508T + 5T^{2} \) |
| 7 | \( 1 + 3.87T + 7T^{2} \) |
| 13 | \( 1 + 0.508T + 13T^{2} \) |
| 17 | \( 1 - 7.36T + 17T^{2} \) |
| 19 | \( 1 + 5.36T + 19T^{2} \) |
| 23 | \( 1 + 2.50T + 23T^{2} \) |
| 29 | \( 1 + 7.36T + 29T^{2} \) |
| 31 | \( 1 - 10.7T + 31T^{2} \) |
| 37 | \( 1 + 8.72T + 37T^{2} \) |
| 41 | \( 1 - 8.37T + 41T^{2} \) |
| 43 | \( 1 - 2.37T + 43T^{2} \) |
| 47 | \( 1 + 5.49T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 - 1.01T + 59T^{2} \) |
| 61 | \( 1 - 7.23T + 61T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 - 14.7T + 73T^{2} \) |
| 79 | \( 1 - 1.14T + 79T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 3.01T + 97T^{2} \) |
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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
−9.273163521100402610255170036629, −8.218707352185089056673963918054, −7.50414077249027763223520972066, −6.62105693040790114387840435168, −5.98186563073511129605729291692, −5.29876786227502208761274261281, −4.05581080728150155661605157838, −3.41934252425097818650714067311, −2.24806773282882522900664640154, −0.59575435269033480206262001868,
0.59575435269033480206262001868, 2.24806773282882522900664640154, 3.41934252425097818650714067311, 4.05581080728150155661605157838, 5.29876786227502208761274261281, 5.98186563073511129605729291692, 6.62105693040790114387840435168, 7.50414077249027763223520972066, 8.218707352185089056673963918054, 9.273163521100402610255170036629