| L(s) = 1 | + (2.92 + 1.21i)2-s + (−1.69 + 0.337i)3-s + (4.27 + 4.27i)4-s + (0.0817 − 0.122i)5-s + (−5.38 − 1.07i)6-s + (−2.35 − 3.51i)7-s + (2.48 + 6.00i)8-s + (2.77 − 1.14i)9-s + (0.387 − 0.259i)10-s + (2.13 − 10.7i)11-s + (−8.71 − 5.82i)12-s + (−12.2 + 12.2i)13-s + (−2.61 − 13.1i)14-s + (−0.0975 + 0.235i)15-s − 3.59i·16-s + (−16.0 + 5.67i)17-s + ⋯ |
| L(s) = 1 | + (1.46 + 0.606i)2-s + (−0.566 + 0.112i)3-s + (1.06 + 1.06i)4-s + (0.0163 − 0.0244i)5-s + (−0.897 − 0.178i)6-s + (−0.335 − 0.502i)7-s + (0.310 + 0.750i)8-s + (0.307 − 0.127i)9-s + (0.0387 − 0.0259i)10-s + (0.194 − 0.975i)11-s + (−0.726 − 0.485i)12-s + (−0.943 + 0.943i)13-s + (−0.187 − 0.940i)14-s + (−0.00650 + 0.0156i)15-s − 0.224i·16-s + (−0.942 + 0.334i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.736 - 0.676i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.74810 + 0.681523i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.74810 + 0.681523i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.69 - 0.337i)T \) |
| 17 | \( 1 + (16.0 - 5.67i)T \) |
| good | 2 | \( 1 + (-2.92 - 1.21i)T + (2.82 + 2.82i)T^{2} \) |
| 5 | \( 1 + (-0.0817 + 0.122i)T + (-9.56 - 23.0i)T^{2} \) |
| 7 | \( 1 + (2.35 + 3.51i)T + (-18.7 + 45.2i)T^{2} \) |
| 11 | \( 1 + (-2.13 + 10.7i)T + (-111. - 46.3i)T^{2} \) |
| 13 | \( 1 + (12.2 - 12.2i)T - 169iT^{2} \) |
| 19 | \( 1 + (-30.9 - 12.8i)T + (255. + 255. i)T^{2} \) |
| 23 | \( 1 + (0.932 + 0.185i)T + (488. + 202. i)T^{2} \) |
| 29 | \( 1 + (27.7 + 18.5i)T + (321. + 776. i)T^{2} \) |
| 31 | \( 1 + (-5.12 - 25.7i)T + (-887. + 367. i)T^{2} \) |
| 37 | \( 1 + (-8.46 + 1.68i)T + (1.26e3 - 523. i)T^{2} \) |
| 41 | \( 1 + (-19.5 - 29.2i)T + (-643. + 1.55e3i)T^{2} \) |
| 43 | \( 1 + (20.6 - 8.56i)T + (1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + (-35.2 + 35.2i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-0.136 - 0.0567i)T + (1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (43.4 + 104. i)T + (-2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (82.1 - 54.8i)T + (1.42e3 - 3.43e3i)T^{2} \) |
| 67 | \( 1 + 24.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-76.3 + 15.1i)T + (4.65e3 - 1.92e3i)T^{2} \) |
| 73 | \( 1 + (-49.1 + 73.5i)T + (-2.03e3 - 4.92e3i)T^{2} \) |
| 79 | \( 1 + (-5.48 + 27.5i)T + (-5.76e3 - 2.38e3i)T^{2} \) |
| 83 | \( 1 + (34.6 - 83.5i)T + (-4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (2.81 + 2.81i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (12.3 + 8.24i)T + (3.60e3 + 8.69e3i)T^{2} \) |
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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
−15.29977762977683495882992648343, −14.13023861307249373712265131768, −13.38561451790116447440313912853, −12.17832984550510155517119382503, −11.21836630423745937703566442865, −9.522375743357430039071654848778, −7.36300117534041176213411409210, −6.28064923187581571447566693072, −5.02305126249802440093090500536, −3.62399163864579090495647342225,
2.64970835423058177095842529679, 4.60708043015662511457837665421, 5.68052005329805302684642971553, 7.19478505089520027898790956720, 9.544408078217463673315863124533, 10.92557248715901617149710459487, 12.07107450811195863302951922566, 12.64609240924918834599679640613, 13.75162443941505877856360394817, 15.01625081569424782242706618070
