| L(s) = 1 | + (−4.44 − 0.254i)2-s + (−1.09 + 3.37i)3-s + (11.7 + 1.34i)4-s + (8.19 + 15.5i)5-s + (5.73 − 14.7i)6-s + (1.62 + 0.687i)7-s + (−16.7 − 2.89i)8-s + (11.6 + 8.45i)9-s + (−32.4 − 71.0i)10-s + (27.6 − 23.7i)11-s + (−17.4 + 38.1i)12-s + (21.2 − 11.9i)13-s + (−7.05 − 3.46i)14-s + (−61.4 + 10.6i)15-s + (−18.4 − 4.28i)16-s + (41.8 − 14.9i)17-s + ⋯ |
| L(s) = 1 | + (−1.57 − 0.0898i)2-s + (−0.211 + 0.650i)3-s + (1.46 + 0.168i)4-s + (0.732 + 1.38i)5-s + (0.390 − 1.00i)6-s + (0.0878 + 0.0371i)7-s + (−0.739 − 0.128i)8-s + (0.431 + 0.313i)9-s + (−1.02 − 2.24i)10-s + (0.758 − 0.651i)11-s + (−0.419 + 0.918i)12-s + (0.452 − 0.255i)13-s + (−0.134 − 0.0661i)14-s + (−1.05 + 0.183i)15-s + (−0.287 − 0.0668i)16-s + (0.597 − 0.213i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.281 - 0.959i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.509025 + 0.679871i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.509025 + 0.679871i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-27.6 + 23.7i)T \) |
| good | 2 | \( 1 + (4.44 + 0.254i)T + (7.94 + 0.911i)T^{2} \) |
| 3 | \( 1 + (1.09 - 3.37i)T + (-21.8 - 15.8i)T^{2} \) |
| 5 | \( 1 + (-8.19 - 15.5i)T + (-70.5 + 103. i)T^{2} \) |
| 7 | \( 1 + (-1.62 - 0.687i)T + (239. + 245. i)T^{2} \) |
| 13 | \( 1 + (-21.2 + 11.9i)T + (1.13e3 - 1.88e3i)T^{2} \) |
| 17 | \( 1 + (-41.8 + 14.9i)T + (3.80e3 - 3.10e3i)T^{2} \) |
| 19 | \( 1 + (3.16 - 110. i)T + (-6.84e3 - 391. i)T^{2} \) |
| 23 | \( 1 + (-122. - 140. i)T + (-1.73e3 + 1.20e4i)T^{2} \) |
| 29 | \( 1 + (127. + 186. i)T + (-8.84e3 + 2.27e4i)T^{2} \) |
| 31 | \( 1 + (1.20 - 5.95i)T + (-2.74e4 - 1.15e4i)T^{2} \) |
| 37 | \( 1 + (45.0 - 41.3i)T + (4.33e3 - 5.04e4i)T^{2} \) |
| 41 | \( 1 + (68.2 + 259. i)T + (-6.00e4 + 3.38e4i)T^{2} \) |
| 43 | \( 1 + (-20.0 - 139. i)T + (-7.62e4 + 2.23e4i)T^{2} \) |
| 47 | \( 1 + (304. - 249. i)T + (2.06e4 - 1.01e5i)T^{2} \) |
| 53 | \( 1 + (-127. + 29.5i)T + (1.33e5 - 6.56e4i)T^{2} \) |
| 59 | \( 1 + (-163. + 622. i)T + (-1.78e5 - 1.00e5i)T^{2} \) |
| 61 | \( 1 + (-105. + 6.02i)T + (2.25e5 - 2.58e4i)T^{2} \) |
| 67 | \( 1 + (523. - 336. i)T + (1.24e5 - 2.73e5i)T^{2} \) |
| 71 | \( 1 + (-244. + 317. i)T + (-9.09e4 - 3.46e5i)T^{2} \) |
| 73 | \( 1 + (-51.1 + 84.9i)T + (-1.81e5 - 3.44e5i)T^{2} \) |
| 79 | \( 1 + (813. - 837. i)T + (-1.40e4 - 4.92e5i)T^{2} \) |
| 83 | \( 1 + (57.5 - 670. i)T + (-5.63e5 - 9.75e4i)T^{2} \) |
| 89 | \( 1 + (-1.33e3 - 393. i)T + (5.93e5 + 3.81e5i)T^{2} \) |
| 97 | \( 1 + (-158. + 299. i)T + (-5.15e5 - 7.53e5i)T^{2} \) |
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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
−13.44369012916059458191108214152, −11.47337889195956628874170451989, −10.88254897744450298135332566004, −9.986265789061979585537752381182, −9.477402198224319516984718110182, −8.015895456396027481528298800105, −6.95758195679278114571544422999, −5.74412667738862205491775413020, −3.45323615462130153446169831913, −1.65274019407750986426629438034,
0.870611689942119471358853122561, 1.69377078897378846643326176640, 4.72958287773727188634384284227, 6.42888778768554261715024969600, 7.32443087587895835902324550508, 8.773225819856995859023572272469, 9.158351868440722466339307445275, 10.21466712194367492467291798474, 11.54319005766282297323337002658, 12.65739887555248710404514166324
