| L(s) = 1 | + (0.642 − 1.26i)2-s + (0.382 − 2.41i)3-s + (−1.17 − 1.61i)4-s + (3.59 + 3.47i)5-s + (−2.79 − 2.03i)6-s + (−7.03 − 7.03i)7-s + (−2.79 + 0.442i)8-s + (2.87 + 0.934i)9-s + (6.68 − 2.29i)10-s + (3.40 + 10.4i)11-s + (−4.35 + 2.21i)12-s + (8.34 + 16.3i)13-s + (−13.3 + 4.34i)14-s + (9.77 − 7.33i)15-s + (−1.23 + 3.80i)16-s + (−2.50 − 15.7i)17-s + ⋯ |
| L(s) = 1 | + (0.321 − 0.630i)2-s + (0.127 − 0.804i)3-s + (−0.293 − 0.404i)4-s + (0.718 + 0.695i)5-s + (−0.466 − 0.338i)6-s + (−1.00 − 1.00i)7-s + (−0.349 + 0.0553i)8-s + (0.319 + 0.103i)9-s + (0.668 − 0.229i)10-s + (0.309 + 0.951i)11-s + (−0.363 + 0.184i)12-s + (0.642 + 1.26i)13-s + (−0.955 + 0.310i)14-s + (0.651 − 0.489i)15-s + (−0.0772 + 0.237i)16-s + (−0.147 − 0.928i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.248 + 0.968i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.248 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.08079 - 0.838421i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.08079 - 0.838421i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.642 + 1.26i)T \) |
| 5 | \( 1 + (-3.59 - 3.47i)T \) |
| good | 3 | \( 1 + (-0.382 + 2.41i)T + (-8.55 - 2.78i)T^{2} \) |
| 7 | \( 1 + (7.03 + 7.03i)T + 49iT^{2} \) |
| 11 | \( 1 + (-3.40 - 10.4i)T + (-97.8 + 71.1i)T^{2} \) |
| 13 | \( 1 + (-8.34 - 16.3i)T + (-99.3 + 136. i)T^{2} \) |
| 17 | \( 1 + (2.50 + 15.7i)T + (-274. + 89.3i)T^{2} \) |
| 19 | \( 1 + (11.0 - 15.2i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (2.76 + 1.41i)T + (310. + 427. i)T^{2} \) |
| 29 | \( 1 + (26.7 + 36.8i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (14.6 + 10.6i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (20.1 - 10.2i)T + (804. - 1.10e3i)T^{2} \) |
| 41 | \( 1 + (6.91 - 21.2i)T + (-1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (-28.9 + 28.9i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (49.6 + 7.86i)T + (2.10e3 + 682. i)T^{2} \) |
| 53 | \( 1 + (-3.99 + 25.2i)T + (-2.67e3 - 868. i)T^{2} \) |
| 59 | \( 1 + (-61.5 - 20.0i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-6.86 - 21.1i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + (-13.1 - 82.8i)T + (-4.26e3 + 1.38e3i)T^{2} \) |
| 71 | \( 1 + (-37.0 + 26.8i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (90.4 + 46.0i)T + (3.13e3 + 4.31e3i)T^{2} \) |
| 79 | \( 1 + (-8.49 - 11.6i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (1.47 - 0.233i)T + (6.55e3 - 2.12e3i)T^{2} \) |
| 89 | \( 1 + (-148. + 48.3i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (-163. - 25.9i)T + (8.94e3 + 2.90e3i)T^{2} \) |
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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
−14.61145731087350583904172705936, −13.58064240016447608687499202152, −13.10233859392262731624126456852, −11.72585092335869343242460005960, −10.29463675046556378637170282705, −9.499640679695146643163201843406, −7.20359818799376567400888764035, −6.40584952979843664456786195580, −3.99733903674374401505735990554, −1.95638220449819752319836750040,
3.48487303843639099610408778590, 5.34168503790708388766700406984, 6.31175416662725605688051840870, 8.608997412382059001694777914697, 9.274561187888797230296294636733, 10.63937202649030678696523845934, 12.67388078986573606818755329274, 13.12223522841762893625748549787, 14.69807591766536455470087769289, 15.74022393914797219376614687602
