L(s) = 1 | + (−0.694 − 3.93i)2-s + (−3.81 + 3.20i)3-s + (−15.0 + 5.47i)4-s + (−0.763 − 0.277i)5-s + (15.2 + 12.8i)6-s + (−52.7 + 91.4i)7-s + (32 + 55.4i)8-s + (−37.8 + 214. i)9-s + (−0.564 + 3.20i)10-s + (145. + 251. i)11-s + (39.8 − 68.9i)12-s + (251. + 211. i)13-s + (396. + 144. i)14-s + (3.80 − 1.38i)15-s + (196. − 164. i)16-s + (102. + 580. i)17-s + ⋯ |
L(s) = 1 | + (−0.122 − 0.696i)2-s + (−0.244 + 0.205i)3-s + (−0.469 + 0.171i)4-s + (−0.0136 − 0.00497i)5-s + (0.173 + 0.145i)6-s + (−0.407 + 0.705i)7-s + (0.176 + 0.306i)8-s + (−0.155 + 0.884i)9-s + (−0.00178 + 0.0101i)10-s + (0.362 + 0.627i)11-s + (0.0798 − 0.138i)12-s + (0.413 + 0.346i)13-s + (0.541 + 0.196i)14-s + (0.00436 − 0.00158i)15-s + (0.191 − 0.160i)16-s + (0.0859 + 0.487i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.707485 + 0.538851i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.707485 + 0.538851i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.694 + 3.93i)T \) |
| 19 | \( 1 + (1.52e3 - 372. i)T \) |
good | 3 | \( 1 + (3.81 - 3.20i)T + (42.1 - 239. i)T^{2} \) |
| 5 | \( 1 + (0.763 + 0.277i)T + (2.39e3 + 2.00e3i)T^{2} \) |
| 7 | \( 1 + (52.7 - 91.4i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-145. - 251. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-251. - 211. i)T + (6.44e4 + 3.65e5i)T^{2} \) |
| 17 | \( 1 + (-102. - 580. i)T + (-1.33e6 + 4.85e5i)T^{2} \) |
| 23 | \( 1 + (1.80e3 - 658. i)T + (4.93e6 - 4.13e6i)T^{2} \) |
| 29 | \( 1 + (-47.2 + 267. i)T + (-1.92e7 - 7.01e6i)T^{2} \) |
| 31 | \( 1 + (-324. + 561. i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 - 5.82e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-3.11e3 + 2.61e3i)T + (2.01e7 - 1.14e8i)T^{2} \) |
| 43 | \( 1 + (-5.14e3 - 1.87e3i)T + (1.12e8 + 9.44e7i)T^{2} \) |
| 47 | \( 1 + (3.53e3 - 2.00e4i)T + (-2.15e8 - 7.84e7i)T^{2} \) |
| 53 | \( 1 + (-3.46e3 + 1.26e3i)T + (3.20e8 - 2.68e8i)T^{2} \) |
| 59 | \( 1 + (-3.98e3 - 2.25e4i)T + (-6.71e8 + 2.44e8i)T^{2} \) |
| 61 | \( 1 + (-3.22e4 + 1.17e4i)T + (6.46e8 - 5.42e8i)T^{2} \) |
| 67 | \( 1 + (-3.58e3 + 2.03e4i)T + (-1.26e9 - 4.61e8i)T^{2} \) |
| 71 | \( 1 + (2.07e4 + 7.56e3i)T + (1.38e9 + 1.15e9i)T^{2} \) |
| 73 | \( 1 + (-6.15e4 + 5.16e4i)T + (3.59e8 - 2.04e9i)T^{2} \) |
| 79 | \( 1 + (5.17e3 - 4.34e3i)T + (5.34e8 - 3.03e9i)T^{2} \) |
| 83 | \( 1 + (3.03e4 - 5.25e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + (3.22e4 + 2.70e4i)T + (9.69e8 + 5.49e9i)T^{2} \) |
| 97 | \( 1 + (-5.91e3 - 3.35e4i)T + (-8.06e9 + 2.93e9i)T^{2} \) |
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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
−15.62201919926337180616663812441, −14.21317676343903245492639448160, −12.90248186184927882144315279499, −11.85572739480245503788222480506, −10.65642752291488011653638441187, −9.481701848303923506657612264221, −8.104415359177344273958594901204, −6.01329837020320853960396526013, −4.24857806486183891145739008470, −2.15460931006822165391361810212,
0.54705690007341072079143643952, 3.81524684042841007601402615805, 5.90258522650577891631034900995, 6.98031927709979572706742815435, 8.537529275748809701766397357002, 9.881369987311827445991794327564, 11.37561071704154475618008257077, 12.85997519596324352394143641685, 13.94154927882629457084353749346, 15.13204420212008615743115842270