L(s) = 1 | + (−0.332 − 1.37i)2-s − 1.47i·3-s + (−1.77 + 0.914i)4-s + (0.707 + 0.707i)5-s + (−2.02 + 0.490i)6-s + (−1.04 − 1.04i)7-s + (1.84 + 2.14i)8-s + 0.828·9-s + (0.736 − 1.20i)10-s + (3.55 + 3.55i)11-s + (1.34 + 2.62i)12-s + (−3.53 − 0.707i)13-s + (−1.08 + 1.77i)14-s + (1.04 − 1.04i)15-s + (2.32 − 3.25i)16-s + 0.171i·17-s + ⋯ |
L(s) = 1 | + (−0.235 − 0.971i)2-s − 0.850i·3-s + (−0.889 + 0.457i)4-s + (0.316 + 0.316i)5-s + (−0.826 + 0.200i)6-s + (−0.393 − 0.393i)7-s + (0.653 + 0.756i)8-s + 0.276·9-s + (0.233 − 0.381i)10-s + (1.07 + 1.07i)11-s + (0.388 + 0.756i)12-s + (−0.980 − 0.196i)13-s + (−0.290 + 0.475i)14-s + (0.269 − 0.269i)15-s + (0.582 − 0.813i)16-s + 0.0416i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0166 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0166 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.516451 - 0.525126i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.516451 - 0.525126i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.332 + 1.37i)T \) |
| 13 | \( 1 + (3.53 + 0.707i)T \) |
good | 3 | \( 1 + 1.47iT - 3T^{2} \) |
| 5 | \( 1 + (-0.707 - 0.707i)T + 5iT^{2} \) |
| 7 | \( 1 + (1.04 + 1.04i)T + 7iT^{2} \) |
| 11 | \( 1 + (-3.55 - 3.55i)T + 11iT^{2} \) |
| 17 | \( 1 - 0.171iT - 17T^{2} \) |
| 19 | \( 1 + (5.03 - 5.03i)T - 19iT^{2} \) |
| 23 | \( 1 + 2.08T + 23T^{2} \) |
| 29 | \( 1 + 3.41T + 29T^{2} \) |
| 31 | \( 1 + (-2.08 + 2.08i)T - 31iT^{2} \) |
| 37 | \( 1 + (6.12 - 6.12i)T - 37iT^{2} \) |
| 41 | \( 1 + (-4.24 - 4.24i)T + 41iT^{2} \) |
| 43 | \( 1 - 5.64T + 43T^{2} \) |
| 47 | \( 1 + (-1.04 - 1.04i)T + 47iT^{2} \) |
| 53 | \( 1 - 8.24T + 53T^{2} \) |
| 59 | \( 1 + (7.11 + 7.11i)T + 59iT^{2} \) |
| 61 | \( 1 + 4.82T + 61T^{2} \) |
| 67 | \( 1 + (-1.47 + 1.47i)T - 67iT^{2} \) |
| 71 | \( 1 + (-9.02 + 9.02i)T - 71iT^{2} \) |
| 73 | \( 1 + (-1.65 + 1.65i)T - 73iT^{2} \) |
| 79 | \( 1 - 0.863iT - 79T^{2} \) |
| 83 | \( 1 + (-0.610 + 0.610i)T - 83iT^{2} \) |
| 89 | \( 1 + (2.24 - 2.24i)T - 89iT^{2} \) |
| 97 | \( 1 + (2.41 + 2.41i)T + 97iT^{2} \) |
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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
−14.81088452988054014489210883287, −13.78869113101720761547275797824, −12.58153092598661185996343032978, −12.10582273298220499241989764899, −10.36922333682052201881202319673, −9.608815193425985099709280348033, −7.86429434946043932685676449256, −6.60772023705902333953368106088, −4.20559786116366499695621368257, −1.99605360711494809008612890403,
4.16638087413090715278607223780, 5.60139926194136483350010858365, 6.99460620215656309404299169721, 8.906165362684874040138849280076, 9.407812936209502821125969029326, 10.77832756973675637671487571085, 12.58585745517888397199281580758, 13.85477749239364507949331055464, 14.93183653317102714105665030623, 15.81182859697575758765889438262