| L(s) = 1 | + 6.08·5-s + 30.4·7-s − 43·11-s − 52·13-s − 85.1·19-s − 42·23-s − 88·25-s + 279.·29-s + 115.·31-s + 185·35-s − 82·37-s + 450.·41-s + 377.·43-s + 74·47-s + 582·49-s + 285.·53-s − 261.·55-s − 20·59-s − 576·61-s − 316.·65-s + 669.·67-s + 1.01e3·71-s + 645·73-s − 1.30e3·77-s − 121.·79-s + 693·83-s + 133.·89-s + ⋯ |
| L(s) = 1 | + 0.544·5-s + 1.64·7-s − 1.17·11-s − 1.10·13-s − 1.02·19-s − 0.380·23-s − 0.703·25-s + 1.79·29-s + 0.669·31-s + 0.893·35-s − 0.364·37-s + 1.71·41-s + 1.33·43-s + 0.229·47-s + 1.69·49-s + 0.740·53-s − 0.641·55-s − 0.0441·59-s − 1.20·61-s − 0.603·65-s + 1.22·67-s + 1.69·71-s + 1.03·73-s − 1.93·77-s − 0.173·79-s + 0.916·83-s + 0.159·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.576105530\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.576105530\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 6.08T + 125T^{2} \) |
| 7 | \( 1 - 30.4T + 343T^{2} \) |
| 11 | \( 1 + 43T + 1.33e3T^{2} \) |
| 13 | \( 1 + 52T + 2.19e3T^{2} \) |
| 17 | \( 1 + 4.91e3T^{2} \) |
| 19 | \( 1 + 85.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 42T + 1.21e4T^{2} \) |
| 29 | \( 1 - 279.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 115.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 82T + 5.06e4T^{2} \) |
| 41 | \( 1 - 450.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 377.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 74T + 1.03e5T^{2} \) |
| 53 | \( 1 - 285.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 20T + 2.05e5T^{2} \) |
| 61 | \( 1 + 576T + 2.26e5T^{2} \) |
| 67 | \( 1 - 669.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.01e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 645T + 3.89e5T^{2} \) |
| 79 | \( 1 + 121.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 693T + 5.71e5T^{2} \) |
| 89 | \( 1 - 133.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.25e3T + 9.12e5T^{2} \) |
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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
−8.868172237499114306141080360758, −7.955286123454932430116244800774, −7.72565926112990690306176361262, −6.51877866852791746816486879131, −5.55416088397044111419674319195, −4.87860792830964178576597640637, −4.23886557731481033638346016729, −2.52249499604647390080798404161, −2.12723139164592059555638474632, −0.75420465872900125380183043880,
0.75420465872900125380183043880, 2.12723139164592059555638474632, 2.52249499604647390080798404161, 4.23886557731481033638346016729, 4.87860792830964178576597640637, 5.55416088397044111419674319195, 6.51877866852791746816486879131, 7.72565926112990690306176361262, 7.955286123454932430116244800774, 8.868172237499114306141080360758
