| L(s) = 1 | + (0.830 − 1.81i)2-s + (2.87 + 0.845i)3-s + (−2.61 − 3.02i)4-s + (−11.8 + 7.61i)5-s + (3.92 − 4.53i)6-s + (−2.71 + 18.8i)7-s + (−7.67 + 2.25i)8-s + (7.57 + 4.86i)9-s + (4.01 + 27.9i)10-s + (10.5 + 23.1i)11-s + (−4.98 − 10.9i)12-s + (7.11 + 49.4i)13-s + (32.1 + 20.6i)14-s + (−40.5 + 11.9i)15-s + (−2.27 + 15.8i)16-s + (11.1 − 12.8i)17-s + ⋯ |
| L(s) = 1 | + (0.293 − 0.643i)2-s + (0.553 + 0.162i)3-s + (−0.327 − 0.377i)4-s + (−1.06 + 0.681i)5-s + (0.267 − 0.308i)6-s + (−0.146 + 1.01i)7-s + (−0.339 + 0.0996i)8-s + (0.280 + 0.180i)9-s + (0.126 + 0.882i)10-s + (0.290 + 0.635i)11-s + (−0.119 − 0.262i)12-s + (0.151 + 1.05i)13-s + (0.612 + 0.393i)14-s + (−0.698 + 0.205i)15-s + (−0.0355 + 0.247i)16-s + (0.159 − 0.183i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.359 - 0.933i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.359 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.21014 + 0.830375i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21014 + 0.830375i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.830 + 1.81i)T \) |
| 3 | \( 1 + (-2.87 - 0.845i)T \) |
| 23 | \( 1 + (94.8 + 56.3i)T \) |
| good | 5 | \( 1 + (11.8 - 7.61i)T + (51.9 - 113. i)T^{2} \) |
| 7 | \( 1 + (2.71 - 18.8i)T + (-329. - 96.6i)T^{2} \) |
| 11 | \( 1 + (-10.5 - 23.1i)T + (-871. + 1.00e3i)T^{2} \) |
| 13 | \( 1 + (-7.11 - 49.4i)T + (-2.10e3 + 618. i)T^{2} \) |
| 17 | \( 1 + (-11.1 + 12.8i)T + (-699. - 4.86e3i)T^{2} \) |
| 19 | \( 1 + (0.505 + 0.582i)T + (-976. + 6.78e3i)T^{2} \) |
| 29 | \( 1 + (32.5 - 37.5i)T + (-3.47e3 - 2.41e4i)T^{2} \) |
| 31 | \( 1 + (35.1 - 10.3i)T + (2.50e4 - 1.61e4i)T^{2} \) |
| 37 | \( 1 + (-179. - 115. i)T + (2.10e4 + 4.60e4i)T^{2} \) |
| 41 | \( 1 + (26.7 - 17.2i)T + (2.86e4 - 6.26e4i)T^{2} \) |
| 43 | \( 1 + (-377. - 110. i)T + (6.68e4 + 4.29e4i)T^{2} \) |
| 47 | \( 1 + 288.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-16.9 + 118. i)T + (-1.42e5 - 4.19e4i)T^{2} \) |
| 59 | \( 1 + (105. + 734. i)T + (-1.97e5 + 5.78e4i)T^{2} \) |
| 61 | \( 1 + (305. - 89.6i)T + (1.90e5 - 1.22e5i)T^{2} \) |
| 67 | \( 1 + (143. - 314. i)T + (-1.96e5 - 2.27e5i)T^{2} \) |
| 71 | \( 1 + (-282. + 619. i)T + (-2.34e5 - 2.70e5i)T^{2} \) |
| 73 | \( 1 + (359. + 414. i)T + (-5.53e4 + 3.85e5i)T^{2} \) |
| 79 | \( 1 + (-152. - 1.05e3i)T + (-4.73e5 + 1.38e5i)T^{2} \) |
| 83 | \( 1 + (-535. - 344. i)T + (2.37e5 + 5.20e5i)T^{2} \) |
| 89 | \( 1 + (-1.03e3 - 304. i)T + (5.93e5 + 3.81e5i)T^{2} \) |
| 97 | \( 1 + (-1.08e3 + 697. i)T + (3.79e5 - 8.30e5i)T^{2} \) |
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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
−12.65626146581830573314913270974, −11.87618074518526797296768268184, −11.11576395959158260158682995623, −9.798306845149533697971688882986, −8.900553579840653463634339870001, −7.68500514681912140460569235171, −6.36833658436227989166078623913, −4.58610050265502696296356692636, −3.48653851582503196864426516652, −2.19063617851287937588651472087,
0.63829133689563548652870938935, 3.47349628474410930283354991625, 4.31302981077679342338861012066, 5.91964797235935560207518530680, 7.47469706131605083575432611545, 7.970961487942315589178324081234, 9.025551827471242274172141217658, 10.43458896183932853888073886365, 11.74496706448415267029784022033, 12.78602082159887276480871657397
