L(s) = 1 | − 3·7-s + 2·11-s + 13-s + 5·17-s − 19-s − 23-s − 5·25-s + 3·29-s − 4·31-s + 2·37-s + 8·41-s + 8·43-s − 8·47-s + 2·49-s − 9·53-s + 59-s + 14·61-s − 13·67-s + 10·71-s + 9·73-s − 6·77-s + 10·79-s + 10·83-s + 12·89-s − 3·91-s + 14·97-s + 14·101-s + ⋯ |
L(s) = 1 | − 1.13·7-s + 0.603·11-s + 0.277·13-s + 1.21·17-s − 0.229·19-s − 0.208·23-s − 25-s + 0.557·29-s − 0.718·31-s + 0.328·37-s + 1.24·41-s + 1.21·43-s − 1.16·47-s + 2/7·49-s − 1.23·53-s + 0.130·59-s + 1.79·61-s − 1.58·67-s + 1.18·71-s + 1.05·73-s − 0.683·77-s + 1.12·79-s + 1.09·83-s + 1.27·89-s − 0.314·91-s + 1.42·97-s + 1.39·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.578550939\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.578550939\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.975493276539202061496102259495, −7.979386204583651359260621889457, −7.36898206685294833870469633157, −6.28626139734514069731905350059, −6.05687642166273662009404713971, −4.93436076241878422688898186043, −3.81643740370862395421698499126, −3.32554545059719619561470562358, −2.13675995078810655105121287439, −0.78177756625263469977425629700,
0.78177756625263469977425629700, 2.13675995078810655105121287439, 3.32554545059719619561470562358, 3.81643740370862395421698499126, 4.93436076241878422688898186043, 6.05687642166273662009404713971, 6.28626139734514069731905350059, 7.36898206685294833870469633157, 7.979386204583651359260621889457, 8.975493276539202061496102259495