L(s) = 1 | + (6.12 + 5.14i)2-s + (−67.0 + 24.3i)3-s + (11.1 + 63.0i)4-s + (−63.2 + 358. i)5-s + (−536. − 195. i)6-s + (−158. − 274. i)7-s + (−256. + 443. i)8-s + (2.22e3 − 1.86e3i)9-s + (−2.23e3 + 1.87e3i)10-s + (1.52e3 − 2.63e3i)11-s + (−2.28e3 − 3.95e3i)12-s + (−3.92e3 − 1.42e3i)13-s + (440. − 2.49e3i)14-s + (−4.50e3 − 2.55e4i)15-s + (−3.84e3 + 1.40e3i)16-s + (−2.29e4 − 1.92e4i)17-s + ⋯ |
L(s) = 1 | + (0.541 + 0.454i)2-s + (−1.43 + 0.521i)3-s + (0.0868 + 0.492i)4-s + (−0.226 + 1.28i)5-s + (−1.01 − 0.368i)6-s + (−0.174 − 0.302i)7-s + (−0.176 + 0.306i)8-s + (1.01 − 0.852i)9-s + (−0.705 + 0.592i)10-s + (0.345 − 0.597i)11-s + (−0.381 − 0.660i)12-s + (−0.494 − 0.180i)13-s + (0.0428 − 0.243i)14-s + (−0.345 − 1.95i)15-s + (−0.234 + 0.0855i)16-s + (−1.13 − 0.952i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.238 + 0.971i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.238 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.119297 - 0.152127i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.119297 - 0.152127i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-6.12 - 5.14i)T \) |
| 19 | \( 1 + (-2.97e4 - 2.42e3i)T \) |
good | 3 | \( 1 + (67.0 - 24.3i)T + (1.67e3 - 1.40e3i)T^{2} \) |
| 5 | \( 1 + (63.2 - 358. i)T + (-7.34e4 - 2.67e4i)T^{2} \) |
| 7 | \( 1 + (158. + 274. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-1.52e3 + 2.63e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (3.92e3 + 1.42e3i)T + (4.80e7 + 4.03e7i)T^{2} \) |
| 17 | \( 1 + (2.29e4 + 1.92e4i)T + (7.12e7 + 4.04e8i)T^{2} \) |
| 23 | \( 1 + (-1.01e4 - 5.75e4i)T + (-3.19e9 + 1.16e9i)T^{2} \) |
| 29 | \( 1 + (7.88e4 - 6.61e4i)T + (2.99e9 - 1.69e10i)T^{2} \) |
| 31 | \( 1 + (1.04e5 + 1.81e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 - 1.57e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + (7.23e5 - 2.63e5i)T + (1.49e11 - 1.25e11i)T^{2} \) |
| 43 | \( 1 + (1.33e5 - 7.57e5i)T + (-2.55e11 - 9.29e10i)T^{2} \) |
| 47 | \( 1 + (4.37e5 - 3.67e5i)T + (8.79e10 - 4.98e11i)T^{2} \) |
| 53 | \( 1 + (2.01e5 + 1.14e6i)T + (-1.10e12 + 4.01e11i)T^{2} \) |
| 59 | \( 1 + (4.79e5 + 4.02e5i)T + (4.32e11 + 2.45e12i)T^{2} \) |
| 61 | \( 1 + (5.50e5 + 3.12e6i)T + (-2.95e12 + 1.07e12i)T^{2} \) |
| 67 | \( 1 + (1.01e6 - 8.48e5i)T + (1.05e12 - 5.96e12i)T^{2} \) |
| 71 | \( 1 + (4.94e5 - 2.80e6i)T + (-8.54e12 - 3.11e12i)T^{2} \) |
| 73 | \( 1 + (4.01e6 - 1.46e6i)T + (8.46e12 - 7.10e12i)T^{2} \) |
| 79 | \( 1 + (-5.63e6 + 2.05e6i)T + (1.47e13 - 1.23e13i)T^{2} \) |
| 83 | \( 1 + (2.92e6 + 5.07e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (1.16e6 + 4.23e5i)T + (3.38e13 + 2.84e13i)T^{2} \) |
| 97 | \( 1 + (1.81e6 + 1.52e6i)T + (1.40e13 + 7.95e13i)T^{2} \) |
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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
−15.70562431752044588799888627594, −14.63970534227204213528622666631, −13.33842028820500533049997656683, −11.55219530214636272629317090228, −11.21051049018709451630821562540, −9.763486293523787915436708612011, −7.30353385810909288275562432850, −6.33990331513798295837353103968, −5.02511236711084448795262216438, −3.37519194203176235225902307776,
0.085736822073696322175387795826, 1.57078154209341050579205143055, 4.48462210898595668282420511186, 5.50002852897610917863447328387, 6.89139499506533559749107147717, 8.965048480835245295092606625302, 10.61013119341797807885147629275, 12.01219320145092313666975231967, 12.33192167811761596658021323218, 13.38099569732020707036152563614