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
−15.13204420212008615743115842270, −13.94154927882629457084353749346, −12.85997519596324352394143641685, −11.37561071704154475618008257077, −9.881369987311827445991794327564, −8.537529275748809701766397357002, −6.98031927709979572706742815435, −5.90258522650577891631034900995, −3.81524684042841007601402615805, −0.54705690007341072079143643952,
2.15460931006822165391361810212, 4.24857806486183891145739008470, 6.01329837020320853960396526013, 8.104415359177344273958594901204, 9.481701848303923506657612264221, 10.65642752291488011653638441187, 11.85572739480245503788222480506, 12.90248186184927882144315279499, 14.21317676343903245492639448160, 15.62201919926337180616663812441