| L(s) = 1 | + (−0.766 + 0.642i)2-s + (−3.00 − 1.09i)3-s + (0.173 − 0.984i)4-s + (−0.0619 − 0.351i)5-s + (3.00 − 1.09i)6-s + (−2.21 + 3.84i)7-s + (0.500 + 0.866i)8-s + (5.53 + 4.64i)9-s + (0.273 + 0.229i)10-s + (−0.5 − 0.866i)11-s + (−1.59 + 2.76i)12-s + (−0.289 + 0.105i)13-s + (−0.770 − 4.36i)14-s + (−0.198 + 1.12i)15-s + (−0.939 − 0.342i)16-s + (1.84 − 1.55i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 + 0.454i)2-s + (−1.73 − 0.631i)3-s + (0.0868 − 0.492i)4-s + (−0.0277 − 0.157i)5-s + (1.22 − 0.446i)6-s + (−0.838 + 1.45i)7-s + (0.176 + 0.306i)8-s + (1.84 + 1.54i)9-s + (0.0864 + 0.0725i)10-s + (−0.150 − 0.261i)11-s + (−0.461 + 0.799i)12-s + (−0.0804 + 0.0292i)13-s + (−0.205 − 1.16i)14-s + (−0.0511 + 0.290i)15-s + (−0.234 − 0.0855i)16-s + (0.448 − 0.376i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.597 + 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.597 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.337938 - 0.169710i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.337938 - 0.169710i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.766 - 0.642i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (3.67 + 2.33i)T \) |
| good | 3 | \( 1 + (3.00 + 1.09i)T + (2.29 + 1.92i)T^{2} \) |
| 5 | \( 1 + (0.0619 + 0.351i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (2.21 - 3.84i)T + (-3.5 - 6.06i)T^{2} \) |
| 13 | \( 1 + (0.289 - 0.105i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.84 + 1.55i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.257 + 1.46i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (2.92 + 2.45i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-4.99 + 8.65i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 6.92T + 37T^{2} \) |
| 41 | \( 1 + (-5.91 - 2.15i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.563 - 3.19i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-8.65 - 7.26i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-1.63 + 9.29i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-9.07 + 7.61i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.671 + 3.80i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (1.19 + 1.00i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.872 - 4.95i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (3.07 + 1.11i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (12.8 + 4.68i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (0.107 - 0.185i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-16.6 + 6.07i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-5.26 + 4.41i)T + (16.8 - 95.5i)T^{2} \) |
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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
−11.20122599336638981455495706870, −10.21076939249237132314753200022, −9.302072838137318608970701777922, −8.234228352414739059569883801485, −7.06210903629018984958310789328, −6.21510801716237107805294276480, −5.73671803491079830758414403895, −4.74108066494816303160381475408, −2.36440752338990473715991329729, −0.46718295829638765568811177089,
1.01058695951399532539521860166, 3.54337260930960874224842685880, 4.37122920028376502649168965239, 5.61739014458354327382308254766, 6.77690151018947272641601741289, 7.26646585990753351353770391557, 8.949469479576326767688049069350, 10.20869480092040543377955876568, 10.35703166573838695464583247096, 10.94558940824171729382076952823
