| L(s) = 1 | + (−2.26 − 1.89i)2-s + (−8.82 + 3.21i)3-s + (0.124 + 0.707i)4-s + (−0.0925 + 0.524i)5-s + (26.0 + 9.48i)6-s + (−11.8 − 20.6i)7-s + (−10.7 + 18.6i)8-s + (46.9 − 39.3i)9-s + (1.20 − 1.01i)10-s + (−18.5 + 32.1i)11-s + (−3.37 − 5.84i)12-s + (−14.9 − 5.45i)13-s + (−12.1 + 69.1i)14-s + (−0.869 − 4.93i)15-s + (65.0 − 23.6i)16-s + (−47.7 − 40.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.799 − 0.671i)2-s + (−1.69 + 0.618i)3-s + (0.0155 + 0.0884i)4-s + (−0.00827 + 0.0469i)5-s + (1.77 + 0.645i)6-s + (−0.642 − 1.11i)7-s + (−0.475 + 0.822i)8-s + (1.73 − 1.45i)9-s + (0.0381 − 0.0319i)10-s + (−0.509 + 0.882i)11-s + (−0.0812 − 0.140i)12-s + (−0.319 − 0.116i)13-s + (−0.232 + 1.32i)14-s + (−0.0149 − 0.0848i)15-s + (1.01 − 0.369i)16-s + (−0.680 − 0.571i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0835i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.00504702 + 0.120655i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00504702 + 0.120655i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (57.8 + 59.2i)T \) |
| good | 2 | \( 1 + (2.26 + 1.89i)T + (1.38 + 7.87i)T^{2} \) |
| 3 | \( 1 + (8.82 - 3.21i)T + (20.6 - 17.3i)T^{2} \) |
| 5 | \( 1 + (0.0925 - 0.524i)T + (-117. - 42.7i)T^{2} \) |
| 7 | \( 1 + (11.8 + 20.6i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (18.5 - 32.1i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (14.9 + 5.45i)T + (1.68e3 + 1.41e3i)T^{2} \) |
| 17 | \( 1 + (47.7 + 40.0i)T + (853. + 4.83e3i)T^{2} \) |
| 23 | \( 1 + (3.18 + 18.0i)T + (-1.14e4 + 4.16e3i)T^{2} \) |
| 29 | \( 1 + (-13.6 + 11.4i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (76.5 + 132. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 41.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + (314. - 114. i)T + (5.27e4 - 4.43e4i)T^{2} \) |
| 43 | \( 1 + (53.2 - 302. i)T + (-7.47e4 - 2.71e4i)T^{2} \) |
| 47 | \( 1 + (-37.2 + 31.2i)T + (1.80e4 - 1.02e5i)T^{2} \) |
| 53 | \( 1 + (84.6 + 479. i)T + (-1.39e5 + 5.09e4i)T^{2} \) |
| 59 | \( 1 + (-160. - 134. i)T + (3.56e4 + 2.02e5i)T^{2} \) |
| 61 | \( 1 + (36.3 + 206. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-327. + 274. i)T + (5.22e4 - 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-105. + 599. i)T + (-3.36e5 - 1.22e5i)T^{2} \) |
| 73 | \( 1 + (299. - 108. i)T + (2.98e5 - 2.50e5i)T^{2} \) |
| 79 | \( 1 + (245. - 89.2i)T + (3.77e5 - 3.16e5i)T^{2} \) |
| 83 | \( 1 + (-328. - 569. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (1.24e3 + 453. i)T + (5.40e5 + 4.53e5i)T^{2} \) |
| 97 | \( 1 + (-245. - 206. i)T + (1.58e5 + 8.98e5i)T^{2} \) |
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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
−17.44575515367664446920581558775, −16.61268111558212563967041994272, −15.20590431164478513249739626974, −12.86979589882644939831673090335, −11.37094169929685193531061925485, −10.47908194312648436096477953346, −9.656771321303331740556373670133, −6.78326650046074924374574843558, −4.84129615331434657141865323334, −0.20860851505968663132450217656,
5.73583334947869435320502790474, 6.79863374639014299483093880549, 8.598247531601124833777252326887, 10.50774860748358818220755699033, 12.11620547045204563056614855016, 12.89944074035852787371913147322, 15.60434269083780497356788312558, 16.46876483109795519269030473839, 17.34449951067221629020400545822, 18.49836212930979654182622307496
