L(s) = 1 | + (1 − 1.73i)2-s + (−3.53 − 6.12i)3-s + (−1.99 − 3.46i)4-s + (−9.89 + 17.1i)5-s − 14.1·6-s − 7.99·8-s + (−11.5 + 19.9i)9-s + (19.7 + 34.2i)10-s + (7 + 12.1i)11-s + (−14.1 + 24.4i)12-s − 50.9·13-s + 140·15-s + (−8 + 13.8i)16-s + (0.707 + 1.22i)17-s + (22.9 + 39.8i)18-s + (−0.707 + 1.22i)19-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.680 − 1.17i)3-s + (−0.249 − 0.433i)4-s + (−0.885 + 1.53i)5-s − 0.962·6-s − 0.353·8-s + (−0.425 + 0.737i)9-s + (0.626 + 1.08i)10-s + (0.191 + 0.332i)11-s + (−0.340 + 0.589i)12-s − 1.08·13-s + 2.40·15-s + (−0.125 + 0.216i)16-s + (0.0100 + 0.0174i)17-s + (0.301 + 0.521i)18-s + (−0.00853 + 0.0147i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0750511 + 0.0917328i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0750511 + 0.0917328i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 + 1.73i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (3.53 + 6.12i)T + (-13.5 + 23.3i)T^{2} \) |
| 5 | \( 1 + (9.89 - 17.1i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-7 - 12.1i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 50.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-0.707 - 1.22i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (0.707 - 1.22i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (70 - 121. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 286T + 2.43e4T^{2} \) |
| 31 | \( 1 + (46.6 + 80.8i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-19 + 32.9i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 125.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 34T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-261. + 453. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-37 - 64.0i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-217. - 375. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-7.07 + 12.2i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (342 + 592. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 588T + 3.57e5T^{2} \) |
| 73 | \( 1 + (135. + 233. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (610 - 1.05e3i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 422.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-309. + 535. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.48e3T + 9.12e5T^{2} \) |
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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
−13.54500556386170296814692520670, −12.38314629547664370236912342926, −11.69373051309877324803011934381, −11.00429238135319684044819983176, −9.769315825724966092190208260970, −7.59455135522075174003361028642, −7.05480315751233636208392848719, −5.76271804229069354324198601339, −3.79113970056088417285437527670, −2.18806336988336164400032956730,
0.06325900694067852553135960071, 3.98747034651942221345603214594, 4.75231984050374488329652596511, 5.67248152748547664248865299410, 7.56303018381291545856212536387, 8.748889376088170182067053094379, 9.701053079281238344731863713559, 11.17656927868192593657787202822, 12.15278416276201060214663877244, 12.93654515551185337009414049631