| L(s) = 1 | + (−0.00957 − 0.0263i)2-s + (2.61 + 0.460i)3-s + (12.2 − 10.2i)4-s + (18.1 + 15.2i)5-s + (−0.0128 − 0.0730i)6-s + (−5.01 − 8.68i)7-s + (−0.775 − 0.447i)8-s + (−69.5 − 25.2i)9-s + (0.226 − 0.622i)10-s + (−47.6 + 82.5i)11-s + (36.7 − 21.2i)12-s + (−184. + 32.4i)13-s + (−0.180 + 0.215i)14-s + (40.3 + 48.0i)15-s + (44.4 − 252. i)16-s + (181. − 66.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.00239 − 0.00657i)2-s + (0.290 + 0.0511i)3-s + (0.766 − 0.642i)4-s + (0.725 + 0.608i)5-s + (−0.000358 − 0.00203i)6-s + (−0.102 − 0.177i)7-s + (−0.0121 − 0.00699i)8-s + (−0.858 − 0.312i)9-s + (0.00226 − 0.00622i)10-s + (−0.393 + 0.681i)11-s + (0.255 − 0.147i)12-s + (−1.08 + 0.192i)13-s + (−0.000921 + 0.00109i)14-s + (0.179 + 0.213i)15-s + (0.173 − 0.984i)16-s + (0.627 − 0.228i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.51360 - 0.0786791i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.51360 - 0.0786791i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (17.9 - 360. i)T \) |
| good | 2 | \( 1 + (0.00957 + 0.0263i)T + (-12.2 + 10.2i)T^{2} \) |
| 3 | \( 1 + (-2.61 - 0.460i)T + (76.1 + 27.7i)T^{2} \) |
| 5 | \( 1 + (-18.1 - 15.2i)T + (108. + 615. i)T^{2} \) |
| 7 | \( 1 + (5.01 + 8.68i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (47.6 - 82.5i)T + (-7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (184. - 32.4i)T + (2.68e4 - 9.76e3i)T^{2} \) |
| 17 | \( 1 + (-181. + 66.0i)T + (6.39e4 - 5.36e4i)T^{2} \) |
| 23 | \( 1 + (-259. + 218. i)T + (4.85e4 - 2.75e5i)T^{2} \) |
| 29 | \( 1 + (140. - 385. i)T + (-5.41e5 - 4.54e5i)T^{2} \) |
| 31 | \( 1 + (640. - 369. i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + 1.24e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-2.79e3 - 492. i)T + (2.65e6 + 9.66e5i)T^{2} \) |
| 43 | \( 1 + (-67.8 - 56.9i)T + (5.93e5 + 3.36e6i)T^{2} \) |
| 47 | \( 1 + (-3.67e3 - 1.33e3i)T + (3.73e6 + 3.13e6i)T^{2} \) |
| 53 | \( 1 + (2.08e3 + 2.47e3i)T + (-1.37e6 + 7.77e6i)T^{2} \) |
| 59 | \( 1 + (-499. - 1.37e3i)T + (-9.28e6 + 7.78e6i)T^{2} \) |
| 61 | \( 1 + (-135. + 113. i)T + (2.40e6 - 1.36e7i)T^{2} \) |
| 67 | \( 1 + (2.41e3 - 6.63e3i)T + (-1.54e7 - 1.29e7i)T^{2} \) |
| 71 | \( 1 + (4.32e3 - 5.15e3i)T + (-4.41e6 - 2.50e7i)T^{2} \) |
| 73 | \( 1 + (-1.49e3 + 8.49e3i)T + (-2.66e7 - 9.71e6i)T^{2} \) |
| 79 | \( 1 + (-9.13e3 - 1.61e3i)T + (3.66e7 + 1.33e7i)T^{2} \) |
| 83 | \( 1 + (3.33e3 + 5.77e3i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (1.04e4 - 1.84e3i)T + (5.89e7 - 2.14e7i)T^{2} \) |
| 97 | \( 1 + (1.44e3 + 3.95e3i)T + (-6.78e7 + 5.69e7i)T^{2} \) |
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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
−17.76403695754639392115690150068, −16.42166610541489132474462864949, −14.74168397606900683244554891008, −14.31794911641014056927875613500, −12.26814676097831064779707604617, −10.66009701938310914708414475179, −9.601685223829933010217189583060, −7.27300378297890609781145448574, −5.72430274772094672022544451364, −2.49631786998259649731046169190,
2.67340281171043244159105154907, 5.63597509524147458666136043189, 7.68836366053115144059647090012, 9.148513136266926680770236219426, 11.04674772812651456611976962225, 12.49328504741117970991066181855, 13.68668981986087082467385017054, 15.27095081766797787683810109699, 16.74646295531366864079098766588, 17.33662431941782475754759629864
