| L(s) = 1 | + 1.75·2-s + 3.47·3-s − 4.93·4-s − 8.23·5-s + 6.08·6-s − 17.9·7-s − 22.6·8-s − 14.9·9-s − 14.4·10-s − 11·11-s − 17.1·12-s − 31.4·14-s − 28.6·15-s − 0.181·16-s + 28.0·17-s − 26.1·18-s − 44.4·19-s + 40.6·20-s − 62.4·21-s − 19.2·22-s − 192.·23-s − 78.7·24-s − 57.2·25-s − 145.·27-s + 88.6·28-s − 49.2·29-s − 50.0·30-s + ⋯ |
| L(s) = 1 | + 0.619·2-s + 0.668·3-s − 0.616·4-s − 0.736·5-s + 0.414·6-s − 0.970·7-s − 1.00·8-s − 0.552·9-s − 0.455·10-s − 0.301·11-s − 0.412·12-s − 0.600·14-s − 0.492·15-s − 0.00284·16-s + 0.400·17-s − 0.342·18-s − 0.536·19-s + 0.454·20-s − 0.648·21-s − 0.186·22-s − 1.74·23-s − 0.669·24-s − 0.458·25-s − 1.03·27-s + 0.598·28-s − 0.315·29-s − 0.304·30-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
| \(L(2)\) |
\(\approx\) |
\(0.6292699891\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6292699891\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 1.75T + 8T^{2} \) |
| 3 | \( 1 - 3.47T + 27T^{2} \) |
| 5 | \( 1 + 8.23T + 125T^{2} \) |
| 7 | \( 1 + 17.9T + 343T^{2} \) |
| 17 | \( 1 - 28.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 44.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 192.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 49.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 118.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 57.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 496.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 226.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 67.8T + 1.03e5T^{2} \) |
| 53 | \( 1 - 428.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 792.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 691.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 700.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 318.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 189.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 894.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 328.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 615.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 752.T + 9.12e5T^{2} \) |
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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
−8.887042480332436844989949905049, −8.068821902293963179362902638783, −7.55568499363074794192130258749, −6.17426829400678972110518796727, −5.77605205766816305466582209802, −4.55489621704203774046737425789, −3.74429286878576939677185006983, −3.27594491431915111221258196068, −2.24551927608281564977673245192, −0.30513791114117953816567002182,
0.30513791114117953816567002182, 2.24551927608281564977673245192, 3.27594491431915111221258196068, 3.74429286878576939677185006983, 4.55489621704203774046737425789, 5.77605205766816305466582209802, 6.17426829400678972110518796727, 7.55568499363074794192130258749, 8.068821902293963179362902638783, 8.887042480332436844989949905049
