| L(s) = 1 | − 2.71i·3-s + 0.0156i·7-s − 4.35·9-s − 6.41·11-s + 2.76i·13-s − 2.07i·17-s + 4.25·19-s + 0.0424·21-s + 5.97i·23-s + 3.67i·27-s − 5.37·29-s + 3.59·31-s + 17.3i·33-s + 9.04i·37-s + 7.48·39-s + ⋯ |
| L(s) = 1 | − 1.56i·3-s + 0.00591i·7-s − 1.45·9-s − 1.93·11-s + 0.765i·13-s − 0.503i·17-s + 0.976·19-s + 0.00926·21-s + 1.24i·23-s + 0.707i·27-s − 0.998·29-s + 0.645·31-s + 3.02i·33-s + 1.48i·37-s + 1.19·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.086908454\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.086908454\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2.71iT - 3T^{2} \) |
| 7 | \( 1 - 0.0156iT - 7T^{2} \) |
| 11 | \( 1 + 6.41T + 11T^{2} \) |
| 13 | \( 1 - 2.76iT - 13T^{2} \) |
| 17 | \( 1 + 2.07iT - 17T^{2} \) |
| 19 | \( 1 - 4.25T + 19T^{2} \) |
| 23 | \( 1 - 5.97iT - 23T^{2} \) |
| 29 | \( 1 + 5.37T + 29T^{2} \) |
| 31 | \( 1 - 3.59T + 31T^{2} \) |
| 37 | \( 1 - 9.04iT - 37T^{2} \) |
| 41 | \( 1 + 7.03T + 41T^{2} \) |
| 43 | \( 1 + 4.96iT - 43T^{2} \) |
| 47 | \( 1 + 5.73iT - 47T^{2} \) |
| 53 | \( 1 - 10.5iT - 53T^{2} \) |
| 59 | \( 1 - 1.13T + 59T^{2} \) |
| 61 | \( 1 - 8.76T + 61T^{2} \) |
| 67 | \( 1 + 11.0iT - 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 - 7.38iT - 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 - 3.44iT - 83T^{2} \) |
| 89 | \( 1 - 9.49T + 89T^{2} \) |
| 97 | \( 1 - 9.21iT - 97T^{2} \) |
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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
−8.074732025343243265064283136266, −7.71001670641496577731026535644, −7.10438894906411783434809379114, −6.44723501377992077935440865674, −5.40896257680392037110426555917, −5.10214286674352807071033965776, −3.60424605129796427064705819963, −2.65733033454136379793378562369, −1.98504007878982449754107269802, −0.906175012772405247850480600087,
0.37408343939491008545430485415, 2.33630697861665353273623081850, 3.10161790044284023524734566302, 3.85428800822567848199330418508, 4.78455855587747458478360357638, 5.32742787850844154239825823883, 5.82946683123750682550523756503, 7.07994321190054923450041597110, 8.003875156362314393232657633812, 8.386706423706746547679962675994
