L(s) = 1 | + (−0.0382 − 0.0661i)5-s + (4.66 + 2.69i)11-s + (4.60 − 2.65i)13-s + 3.78·17-s − 5.01i·19-s + (−2.02 + 1.16i)23-s + (2.49 − 4.32i)25-s + (−8.84 − 5.10i)29-s + (4.97 − 2.87i)31-s − 0.708·37-s + (3.29 + 5.71i)41-s + (0.716 − 1.24i)43-s + (−1.46 + 2.53i)47-s − 12.1i·53-s − 0.411i·55-s + ⋯ |
L(s) = 1 | + (−0.0170 − 0.0295i)5-s + (1.40 + 0.811i)11-s + (1.27 − 0.737i)13-s + 0.917·17-s − 1.14i·19-s + (−0.422 + 0.243i)23-s + (0.499 − 0.865i)25-s + (−1.64 − 0.948i)29-s + (0.893 − 0.516i)31-s − 0.116·37-s + (0.515 + 0.892i)41-s + (0.109 − 0.189i)43-s + (−0.213 + 0.369i)47-s − 1.66i·53-s − 0.0554i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 + 0.647i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.762 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.351495873\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.351495873\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.0382 + 0.0661i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-4.66 - 2.69i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.60 + 2.65i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 3.78T + 17T^{2} \) |
| 19 | \( 1 + 5.01iT - 19T^{2} \) |
| 23 | \( 1 + (2.02 - 1.16i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (8.84 + 5.10i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.97 + 2.87i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 0.708T + 37T^{2} \) |
| 41 | \( 1 + (-3.29 - 5.71i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.716 + 1.24i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.46 - 2.53i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 12.1iT - 53T^{2} \) |
| 59 | \( 1 + (0.289 + 0.502i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.40 + 1.38i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.63 + 4.56i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 3.32iT - 71T^{2} \) |
| 73 | \( 1 - 7.12iT - 73T^{2} \) |
| 79 | \( 1 + (0.469 - 0.812i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (6.49 - 11.2i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 3.03T + 89T^{2} \) |
| 97 | \( 1 + (6.18 + 3.56i)T + (48.5 + 84.0i)T^{2} \) |
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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
−8.127737205495906890001278032099, −7.43196359253513746055552744287, −6.56788407030990709862597287592, −6.09163169230837991289732316894, −5.23419248057651134472347178144, −4.26663994482285856286612857631, −3.74753948604036406162549896238, −2.77119498262484040816888055033, −1.66098003777611759730792610932, −0.73084941229013101757483295929,
1.14545396999576397336107769666, 1.68784478326644123338110931358, 3.27474932833054775218350834614, 3.66089195456943223307269653438, 4.42501145325039294388601460184, 5.70343672370490021923972216160, 5.96958453118929446888704097836, 6.80905305246126360290545794035, 7.52793132673350320327283777053, 8.398665839867747260710922149970