L(s) = 1 | + (−1.37 − 0.332i)2-s + 1.47i·3-s + (1.77 + 0.914i)4-s + (0.707 + 0.707i)5-s + (0.490 − 2.02i)6-s + (1.04 + 1.04i)7-s + (−2.14 − 1.84i)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 | + (−0.971 − 0.235i)2-s + 0.850i·3-s + (0.889 + 0.457i)4-s + (0.316 + 0.316i)5-s + (0.200 − 0.826i)6-s + (0.393 + 0.393i)7-s + (−0.756 − 0.653i)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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 - 0.568i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.580337 + 0.181002i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.580337 + 0.181002i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.37 + 0.332i)T \) |
| 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
−15.77169893888192543984164944328, −14.91049644126229172027590644880, −13.25124367323121791021311445650, −11.70654316803090544953144988970, −10.63715670884158388629471046477, −9.818665056297153844284809077893, −8.617027492698527973125131110966, −7.21484705843595877106643089991, −5.27537125655945897425751179119, −2.90441970556398063376289308610,
1.86921445022426664662578267791, 5.33902559114650045858671359001, 7.25266987496594696569803279340, 7.67612274844376929891940654308, 9.454981696008928532702650124557, 10.41068669110292873539074336885, 11.97162463481887189292741635736, 12.94980312721025985092375804476, 14.36706003906047866157773298731, 15.54634689402109365929981718497