L(s) = 1 | + (1.56 − 2.70i)2-s + (0.268 − 0.465i)3-s + (−0.889 − 1.53i)4-s − 0.516·5-s + (−0.840 − 1.45i)6-s + (17.3 + 30.0i)7-s + 19.4·8-s + (13.3 + 23.1i)9-s + (−0.807 + 1.39i)10-s + (−5.5 + 9.52i)11-s − 0.955·12-s + (−18.8 − 42.9i)13-s + 108.·14-s + (−0.138 + 0.240i)15-s + (37.5 − 65.0i)16-s + (−57.0 − 98.8i)17-s + ⋯ |
L(s) = 1 | + (0.552 − 0.957i)2-s + (0.0517 − 0.0896i)3-s + (−0.111 − 0.192i)4-s − 0.0461·5-s + (−0.0571 − 0.0990i)6-s + (0.935 + 1.61i)7-s + 0.859·8-s + (0.494 + 0.856i)9-s + (−0.0255 + 0.0442i)10-s + (−0.150 + 0.261i)11-s − 0.0229·12-s + (−0.401 − 0.915i)13-s + 2.06·14-s + (−0.00238 + 0.00413i)15-s + (0.586 − 1.01i)16-s + (−0.814 − 1.41i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.57077 - 0.492861i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.57077 - 0.492861i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (5.5 - 9.52i)T \) |
| 13 | \( 1 + (18.8 + 42.9i)T \) |
good | 2 | \( 1 + (-1.56 + 2.70i)T + (-4 - 6.92i)T^{2} \) |
| 3 | \( 1 + (-0.268 + 0.465i)T + (-13.5 - 23.3i)T^{2} \) |
| 5 | \( 1 + 0.516T + 125T^{2} \) |
| 7 | \( 1 + (-17.3 - 30.0i)T + (-171.5 + 297. i)T^{2} \) |
| 17 | \( 1 + (57.0 + 98.8i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-61.4 - 106. i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-33.4 + 57.9i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (32.3 - 56.0i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 - 238.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (14.1 - 24.5i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-154. + 266. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (123. + 213. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + 165.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 79.9T + 1.48e5T^{2} \) |
| 59 | \( 1 + (133. + 231. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (175. + 304. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (307. - 533. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (235. + 407. i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + 65.0T + 3.89e5T^{2} \) |
| 79 | \( 1 - 982.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.03e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-657. + 1.13e3i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (224. + 388. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−12.28907987417103638449074806279, −11.84942734140569528812684861004, −10.83319103517526911201327622254, −9.764319045708891392963030126844, −8.296833291881157485868397430763, −7.47079689090819306824407032669, −5.43545244929508786236921158926, −4.67117321816986232376377854708, −2.80292832006138755169756943800, −1.89586322612304206692715115392,
1.31203261468935639430208263668, 4.05399907890437417543221148518, 4.69413112369163487131907043475, 6.35997776002852977206187626038, 7.15320847498529912126774894283, 8.062753610895631434622890875428, 9.642457144610103777660949430809, 10.74026276777185979161967444330, 11.59287158926302088872460370377, 13.30157333902721194969179429019