| L(s) = 1 | + (5.50 + 1.47i)2-s + (−4.28 − 7.41i)3-s + (14.3 + 8.26i)4-s + (−29.3 + 29.3i)5-s + (−12.6 − 47.1i)6-s + (45.8 − 12.2i)7-s + (2.12 + 2.12i)8-s + (3.84 − 6.66i)9-s + (−204. + 118. i)10-s + (11.4 − 42.7i)11-s − 141. i·12-s + (102. + 134. i)13-s + 270.·14-s + (343. + 91.9i)15-s + (−123. − 214. i)16-s + (−205. − 118. i)17-s + ⋯ |
| L(s) = 1 | + (1.37 + 0.369i)2-s + (−0.475 − 0.823i)3-s + (0.894 + 0.516i)4-s + (−1.17 + 1.17i)5-s + (−0.351 − 1.31i)6-s + (0.935 − 0.250i)7-s + (0.0332 + 0.0332i)8-s + (0.0475 − 0.0823i)9-s + (−2.04 + 1.18i)10-s + (0.0946 − 0.353i)11-s − 0.982i·12-s + (0.605 + 0.795i)13-s + 1.38·14-s + (1.52 + 0.408i)15-s + (−0.482 − 0.836i)16-s + (−0.711 − 0.410i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.65990 + 0.0837813i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.65990 + 0.0837813i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-102. - 134. i)T \) |
| good | 2 | \( 1 + (-5.50 - 1.47i)T + (13.8 + 8i)T^{2} \) |
| 3 | \( 1 + (4.28 + 7.41i)T + (-40.5 + 70.1i)T^{2} \) |
| 5 | \( 1 + (29.3 - 29.3i)T - 625iT^{2} \) |
| 7 | \( 1 + (-45.8 + 12.2i)T + (2.07e3 - 1.20e3i)T^{2} \) |
| 11 | \( 1 + (-11.4 + 42.7i)T + (-1.26e4 - 7.32e3i)T^{2} \) |
| 17 | \( 1 + (205. + 118. i)T + (4.17e4 + 7.23e4i)T^{2} \) |
| 19 | \( 1 + (-127. - 476. i)T + (-1.12e5 + 6.51e4i)T^{2} \) |
| 23 | \( 1 + (258. - 149. i)T + (1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + (-185. - 321. i)T + (-3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 + (-625. + 625. i)T - 9.23e5iT^{2} \) |
| 37 | \( 1 + (-197. + 737. i)T + (-1.62e6 - 9.37e5i)T^{2} \) |
| 41 | \( 1 + (671. + 179. i)T + (2.44e6 + 1.41e6i)T^{2} \) |
| 43 | \( 1 + (-975. - 563. i)T + (1.70e6 + 2.96e6i)T^{2} \) |
| 47 | \( 1 + (-609. - 609. i)T + 4.87e6iT^{2} \) |
| 53 | \( 1 - 897.T + 7.89e6T^{2} \) |
| 59 | \( 1 + (-480. + 128. i)T + (1.04e7 - 6.05e6i)T^{2} \) |
| 61 | \( 1 + (124. - 215. i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-1.80e3 - 483. i)T + (1.74e7 + 1.00e7i)T^{2} \) |
| 71 | \( 1 + (-776. - 2.89e3i)T + (-2.20e7 + 1.27e7i)T^{2} \) |
| 73 | \( 1 + (6.81e3 + 6.81e3i)T + 2.83e7iT^{2} \) |
| 79 | \( 1 - 2.50e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + (-7.97e3 + 7.97e3i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 + (3.75e3 - 1.40e4i)T + (-5.43e7 - 3.13e7i)T^{2} \) |
| 97 | \( 1 + (1.91e3 + 7.15e3i)T + (-7.66e7 + 4.42e7i)T^{2} \) |
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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
−18.95258277289422147584384739615, −18.07940775604753290433129974986, −15.99458961273522623666368085450, −14.75604539432826648476450321218, −13.78894193194284409114416631372, −12.03996313930667161738317240524, −11.28885408821161761358585395848, −7.55246064454937909715738361944, −6.33768271879002592537011020582, −3.95818339670424727662495914871,
4.27379297780314554209646544886, 5.10945015482522752527988371318, 8.437406892107221763961920381330, 11.03987864072151490560879448588, 11.99425421088550294031774996499, 13.29594821865098378244885757131, 15.22100009072340386880936576213, 15.83323541669197278185278872606, 17.52530044174192023585597087282, 19.90330862305989271587951651085
