| L(s) = 1 | + (1.72 + 0.199i)3-s − 3.94·5-s + (−0.792 − 2.52i)7-s + (2.92 + 0.686i)9-s + i·11-s + 0.0683i·13-s + (−6.78 − 0.786i)15-s − 3.76·17-s − 1.51i·19-s + (−0.860 − 4.50i)21-s + 7.42i·23-s + 10.5·25-s + (4.88 + 1.76i)27-s + 6.29i·29-s + 7.71i·31-s + ⋯ |
| L(s) = 1 | + (0.993 + 0.115i)3-s − 1.76·5-s + (−0.299 − 0.954i)7-s + (0.973 + 0.228i)9-s + 0.301i·11-s + 0.0189i·13-s + (−1.75 − 0.203i)15-s − 0.913·17-s − 0.346i·19-s + (−0.187 − 0.982i)21-s + 1.54i·23-s + 2.10·25-s + (0.940 + 0.339i)27-s + 1.16i·29-s + 1.38i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.001878549\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.001878549\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.72 - 0.199i)T \) |
| 7 | \( 1 + (0.792 + 2.52i)T \) |
| 11 | \( 1 - iT \) |
| good | 5 | \( 1 + 3.94T + 5T^{2} \) |
| 13 | \( 1 - 0.0683iT - 13T^{2} \) |
| 17 | \( 1 + 3.76T + 17T^{2} \) |
| 19 | \( 1 + 1.51iT - 19T^{2} \) |
| 23 | \( 1 - 7.42iT - 23T^{2} \) |
| 29 | \( 1 - 6.29iT - 29T^{2} \) |
| 31 | \( 1 - 7.71iT - 31T^{2} \) |
| 37 | \( 1 + 11.8T + 37T^{2} \) |
| 41 | \( 1 - 3.60T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 - 3.21T + 47T^{2} \) |
| 53 | \( 1 - 3.75iT - 53T^{2} \) |
| 59 | \( 1 + 2.17T + 59T^{2} \) |
| 61 | \( 1 + 1.89iT - 61T^{2} \) |
| 67 | \( 1 + 2.06T + 67T^{2} \) |
| 71 | \( 1 - 2.38iT - 71T^{2} \) |
| 73 | \( 1 - 9.38iT - 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 + 7.60T + 83T^{2} \) |
| 89 | \( 1 + 1.20T + 89T^{2} \) |
| 97 | \( 1 + 11.9iT - 97T^{2} \) |
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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
−9.146793356793868327141952075690, −8.739894904766870738272865345899, −7.76507720482424891142387880879, −7.24172032746253146578411877284, −6.85474905150717028598207350250, −5.06256376421298585731484640417, −4.20574577693474153332854674041, −3.68499747983980404525799105334, −2.92323893211856019712224433598, −1.30853252313058567708838251869,
0.34352258004875938087068996536, 2.24361700109908763077442269221, 3.03933607990387005328189470775, 4.02161357389337163769044327130, 4.50626408854743456521787757873, 5.94205796295607561477206312620, 6.88646625507231318457421782029, 7.62385987739919628487443668507, 8.392159929250855795796643933398, 8.705435244042339621380039170765
