| L(s) = 1 | + (1.68 − 1.08i)2-s + (0.426 − 2.96i)3-s + (1.66 − 3.63i)4-s + (−20.3 + 5.97i)5-s + (−2.49 − 5.45i)6-s + (5.13 + 5.92i)7-s + (−1.13 − 7.91i)8-s + (−8.63 − 2.53i)9-s + (−27.7 + 32.0i)10-s + (−33.5 − 21.5i)11-s + (−10.0 − 6.48i)12-s + (−36.4 + 42.1i)13-s + (15.0 + 4.41i)14-s + (9.06 + 63.0i)15-s + (−10.4 − 12.0i)16-s + (−37.7 − 82.7i)17-s + ⋯ |
| L(s) = 1 | + (0.594 − 0.382i)2-s + (0.0821 − 0.571i)3-s + (0.207 − 0.454i)4-s + (−1.82 + 0.534i)5-s + (−0.169 − 0.371i)6-s + (0.277 + 0.319i)7-s + (−0.0503 − 0.349i)8-s + (−0.319 − 0.0939i)9-s + (−0.878 + 1.01i)10-s + (−0.920 − 0.591i)11-s + (−0.242 − 0.156i)12-s + (−0.778 + 0.898i)13-s + (0.287 + 0.0843i)14-s + (0.155 + 1.08i)15-s + (−0.163 − 0.188i)16-s + (−0.538 − 1.17i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0575885 + 0.202960i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0575885 + 0.202960i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.68 + 1.08i)T \) |
| 3 | \( 1 + (-0.426 + 2.96i)T \) |
| 23 | \( 1 + (-107. - 24.8i)T \) |
| good | 5 | \( 1 + (20.3 - 5.97i)T + (105. - 67.5i)T^{2} \) |
| 7 | \( 1 + (-5.13 - 5.92i)T + (-48.8 + 339. i)T^{2} \) |
| 11 | \( 1 + (33.5 + 21.5i)T + (552. + 1.21e3i)T^{2} \) |
| 13 | \( 1 + (36.4 - 42.1i)T + (-312. - 2.17e3i)T^{2} \) |
| 17 | \( 1 + (37.7 + 82.7i)T + (-3.21e3 + 3.71e3i)T^{2} \) |
| 19 | \( 1 + (47.3 - 103. i)T + (-4.49e3 - 5.18e3i)T^{2} \) |
| 29 | \( 1 + (62.5 + 136. i)T + (-1.59e4 + 1.84e4i)T^{2} \) |
| 31 | \( 1 + (-0.528 - 3.67i)T + (-2.85e4 + 8.39e3i)T^{2} \) |
| 37 | \( 1 + (210. + 61.7i)T + (4.26e4 + 2.73e4i)T^{2} \) |
| 41 | \( 1 + (136. - 40.2i)T + (5.79e4 - 3.72e4i)T^{2} \) |
| 43 | \( 1 + (-52.5 + 365. i)T + (-7.62e4 - 2.23e4i)T^{2} \) |
| 47 | \( 1 + 145.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (84.5 + 97.5i)T + (-2.11e4 + 1.47e5i)T^{2} \) |
| 59 | \( 1 + (-328. + 378. i)T + (-2.92e4 - 2.03e5i)T^{2} \) |
| 61 | \( 1 + (-117. - 817. i)T + (-2.17e5 + 6.39e4i)T^{2} \) |
| 67 | \( 1 + (384. - 247. i)T + (1.24e5 - 2.73e5i)T^{2} \) |
| 71 | \( 1 + (109. - 70.0i)T + (1.48e5 - 3.25e5i)T^{2} \) |
| 73 | \( 1 + (-102. + 223. i)T + (-2.54e5 - 2.93e5i)T^{2} \) |
| 79 | \( 1 + (178. - 206. i)T + (-7.01e4 - 4.88e5i)T^{2} \) |
| 83 | \( 1 + (368. + 108. i)T + (4.81e5 + 3.09e5i)T^{2} \) |
| 89 | \( 1 + (183. - 1.27e3i)T + (-6.76e5 - 1.98e5i)T^{2} \) |
| 97 | \( 1 + (-1.19e3 + 350. i)T + (7.67e5 - 4.93e5i)T^{2} \) |
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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
−11.83511416292065743798744660711, −11.59960727072399069310666417209, −10.48261800788519088442150503053, −8.717139259063596258309846693930, −7.64463923853333261918113034642, −6.83117283895075204447321918104, −5.11147352644500313241628173145, −3.79332750589844060230816248196, −2.51307792428551577871294504825, −0.081640652716098151777602393903,
3.13915444543484027096367420853, 4.46450011230402726580732351256, 5.00743271956221475290354538252, 7.08034917987205496167848190047, 7.935428364112716395474157600337, 8.783240119625262588360066003470, 10.58062103054557675953802275897, 11.27792126712632794537229298717, 12.57780516542054164114132465021, 12.97939892032731766132411654035
