| L(s) = 1 | − 4.34·2-s − 6.31·3-s + 10.9·4-s + 2.31·5-s + 27.4·6-s + 33.8·7-s − 12.6·8-s + 12.8·9-s − 10.0·10-s − 11·11-s − 68.8·12-s − 147.·14-s − 14.6·15-s − 32.1·16-s − 11.5·17-s − 55.8·18-s − 31.6·19-s + 25.3·20-s − 213.·21-s + 47.8·22-s + 31.7·23-s + 79.9·24-s − 119.·25-s + 89.3·27-s + 369.·28-s − 263.·29-s + 63.6·30-s + ⋯ |
| L(s) = 1 | − 1.53·2-s − 1.21·3-s + 1.36·4-s + 0.207·5-s + 1.86·6-s + 1.82·7-s − 0.560·8-s + 0.475·9-s − 0.318·10-s − 0.301·11-s − 1.65·12-s − 2.81·14-s − 0.251·15-s − 0.503·16-s − 0.164·17-s − 0.731·18-s − 0.381·19-s + 0.282·20-s − 2.22·21-s + 0.463·22-s + 0.288·23-s + 0.680·24-s − 0.956·25-s + 0.637·27-s + 2.49·28-s − 1.68·29-s + 0.387·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5186354784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5186354784\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 4.34T + 8T^{2} \) |
| 3 | \( 1 + 6.31T + 27T^{2} \) |
| 5 | \( 1 - 2.31T + 125T^{2} \) |
| 7 | \( 1 - 33.8T + 343T^{2} \) |
| 17 | \( 1 + 11.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 31.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 31.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 263.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 320.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 105.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 213.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 145.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 246.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 24.9T + 1.48e5T^{2} \) |
| 59 | \( 1 + 288.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 556.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 542.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 845.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 451.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 591.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 973.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 959.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 955.T + 9.12e5T^{2} \) |
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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
−8.979407734000296635553867352663, −7.933455009157419662590086036972, −7.68694944315746772567346870706, −6.69061300492466369503174999292, −5.66051351104275415066494520954, −5.11090679343139898976560297667, −4.11787904254384056809979881724, −2.18027186774178930252586353122, −1.54486634712596821598820976550, −0.47295498888752633896356475846,
0.47295498888752633896356475846, 1.54486634712596821598820976550, 2.18027186774178930252586353122, 4.11787904254384056809979881724, 5.11090679343139898976560297667, 5.66051351104275415066494520954, 6.69061300492466369503174999292, 7.68694944315746772567346870706, 7.933455009157419662590086036972, 8.979407734000296635553867352663
