L(s) = 1 | + 0.0785·3-s + 5·5-s + 6.13·7-s − 26.9·9-s − 39.3·11-s − 8.89·13-s + 0.392·15-s − 48.1·17-s + 43.4·19-s + 0.481·21-s − 23·23-s + 25·25-s − 4.23·27-s − 265.·29-s + 71.4·31-s − 3.08·33-s + 30.6·35-s + 54.2·37-s − 0.698·39-s + 139.·41-s + 298.·43-s − 134.·45-s + 235.·47-s − 305.·49-s − 3.77·51-s + 619.·53-s − 196.·55-s + ⋯ |
L(s) = 1 | + 0.0151·3-s + 0.447·5-s + 0.331·7-s − 0.999·9-s − 1.07·11-s − 0.189·13-s + 0.00675·15-s − 0.686·17-s + 0.524·19-s + 0.00500·21-s − 0.208·23-s + 0.200·25-s − 0.0302·27-s − 1.69·29-s + 0.414·31-s − 0.0162·33-s + 0.148·35-s + 0.240·37-s − 0.00286·39-s + 0.530·41-s + 1.05·43-s − 0.447·45-s + 0.729·47-s − 0.890·49-s − 0.0103·51-s + 1.60·53-s − 0.482·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.642985966\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.642985966\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 + 23T \) |
good | 3 | \( 1 - 0.0785T + 27T^{2} \) |
| 7 | \( 1 - 6.13T + 343T^{2} \) |
| 11 | \( 1 + 39.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 8.89T + 2.19e3T^{2} \) |
| 17 | \( 1 + 48.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 43.4T + 6.85e3T^{2} \) |
| 29 | \( 1 + 265.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 71.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 54.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 139.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 298.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 235.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 619.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 66.8T + 2.05e5T^{2} \) |
| 61 | \( 1 + 206.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 653.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 641.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.12e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.17e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.50e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.65e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 501.T + 9.12e5T^{2} \) |
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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
−8.978132779219305518747952353195, −8.028463973280273024349832116115, −7.50036783070794309356878314607, −6.38774988176374613307225536798, −5.56361624508310314573316906998, −5.05267723617911910616782628432, −3.87452172003777083667304973368, −2.72549893785988672425757666563, −2.07930434766038296564802902816, −0.57332705008364078458165583906,
0.57332705008364078458165583906, 2.07930434766038296564802902816, 2.72549893785988672425757666563, 3.87452172003777083667304973368, 5.05267723617911910616782628432, 5.56361624508310314573316906998, 6.38774988176374613307225536798, 7.50036783070794309356878314607, 8.028463973280273024349832116115, 8.978132779219305518747952353195