| L(s) = 1 | + 1.44·3-s − 2.99·5-s − 1.69·7-s − 0.900·9-s + 5.69·11-s − 6.47·13-s − 4.34·15-s + 5.94·17-s + 0.280·19-s − 2.45·21-s + 3.98·25-s − 5.65·27-s + 1.20·29-s − 2.99·31-s + 8.25·33-s + 5.07·35-s + 0.265·37-s − 9.37·39-s + 5.81·41-s − 5.66·43-s + 2.69·45-s − 1.51·47-s − 4.13·49-s + 8.61·51-s + 6.60·53-s − 17.0·55-s + 0.405·57-s + ⋯ |
| L(s) = 1 | + 0.836·3-s − 1.34·5-s − 0.639·7-s − 0.300·9-s + 1.71·11-s − 1.79·13-s − 1.12·15-s + 1.44·17-s + 0.0642·19-s − 0.535·21-s + 0.796·25-s − 1.08·27-s + 0.223·29-s − 0.537·31-s + 1.43·33-s + 0.857·35-s + 0.0436·37-s − 1.50·39-s + 0.908·41-s − 0.863·43-s + 0.402·45-s − 0.221·47-s − 0.590·49-s + 1.20·51-s + 0.907·53-s − 2.30·55-s + 0.0537·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.533852717\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.533852717\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 1.44T + 3T^{2} \) |
| 5 | \( 1 + 2.99T + 5T^{2} \) |
| 7 | \( 1 + 1.69T + 7T^{2} \) |
| 11 | \( 1 - 5.69T + 11T^{2} \) |
| 13 | \( 1 + 6.47T + 13T^{2} \) |
| 17 | \( 1 - 5.94T + 17T^{2} \) |
| 19 | \( 1 - 0.280T + 19T^{2} \) |
| 29 | \( 1 - 1.20T + 29T^{2} \) |
| 31 | \( 1 + 2.99T + 31T^{2} \) |
| 37 | \( 1 - 0.265T + 37T^{2} \) |
| 41 | \( 1 - 5.81T + 41T^{2} \) |
| 43 | \( 1 + 5.66T + 43T^{2} \) |
| 47 | \( 1 + 1.51T + 47T^{2} \) |
| 53 | \( 1 - 6.60T + 53T^{2} \) |
| 59 | \( 1 - 3.21T + 59T^{2} \) |
| 61 | \( 1 - 8.76T + 61T^{2} \) |
| 67 | \( 1 + 3.11T + 67T^{2} \) |
| 71 | \( 1 + 2.19T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 - 0.653T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 + 9.80T + 89T^{2} \) |
| 97 | \( 1 - 3.82T + 97T^{2} \) |
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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
−8.337361395615347451032436808523, −7.64859365122773412419545777875, −7.20701737487029932479680803133, −6.38527343606962976249082590399, −5.35302880726463571026080586795, −4.39524022491043398188922815307, −3.57045705107386464644399441420, −3.24199192267806286050359259076, −2.13417142182077879097786752978, −0.65536906326806390753875862888,
0.65536906326806390753875862888, 2.13417142182077879097786752978, 3.24199192267806286050359259076, 3.57045705107386464644399441420, 4.39524022491043398188922815307, 5.35302880726463571026080586795, 6.38527343606962976249082590399, 7.20701737487029932479680803133, 7.64859365122773412419545777875, 8.337361395615347451032436808523
