L(s) = 1 | + (0.995 − 0.0950i)2-s + (−1.65 + 1.06i)3-s + (0.981 − 0.189i)4-s + (−0.142 + 0.989i)5-s + (−1.54 + 1.21i)6-s + (1.03 + 1.44i)7-s + (0.959 − 0.281i)8-s + (1.18 − 2.59i)9-s + (−0.0475 + 0.998i)10-s + (−1.42 + 1.35i)12-s + (1.16 + 1.34i)14-s + (−0.815 − 1.78i)15-s + (0.928 − 0.371i)16-s + (0.934 − 2.69i)18-s + (0.0475 + 0.998i)20-s + (−3.24 − 1.29i)21-s + ⋯ |
L(s) = 1 | + (0.995 − 0.0950i)2-s + (−1.65 + 1.06i)3-s + (0.981 − 0.189i)4-s + (−0.142 + 0.989i)5-s + (−1.54 + 1.21i)6-s + (1.03 + 1.44i)7-s + (0.959 − 0.281i)8-s + (1.18 − 2.59i)9-s + (−0.0475 + 0.998i)10-s + (−1.42 + 1.35i)12-s + (1.16 + 1.34i)14-s + (−0.815 − 1.78i)15-s + (0.928 − 0.371i)16-s + (0.934 − 2.69i)18-s + (0.0475 + 0.998i)20-s + (−3.24 − 1.29i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.233 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.233 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.308988016\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.308988016\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.995 + 0.0950i)T \) |
| 5 | \( 1 + (0.142 - 0.989i)T \) |
| 67 | \( 1 + (-0.327 + 0.945i)T \) |
good | 3 | \( 1 + (1.65 - 1.06i)T + (0.415 - 0.909i)T^{2} \) |
| 7 | \( 1 + (-1.03 - 1.44i)T + (-0.327 + 0.945i)T^{2} \) |
| 11 | \( 1 + (-0.235 - 0.971i)T^{2} \) |
| 13 | \( 1 + (0.888 - 0.458i)T^{2} \) |
| 17 | \( 1 + (-0.928 - 0.371i)T^{2} \) |
| 19 | \( 1 + (0.327 + 0.945i)T^{2} \) |
| 23 | \( 1 + (0.888 + 0.458i)T + (0.580 + 0.814i)T^{2} \) |
| 29 | \( 1 + (0.0475 - 0.0824i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.888 + 0.458i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.154 + 0.445i)T + (-0.786 + 0.618i)T^{2} \) |
| 43 | \( 1 + (0.428 - 0.494i)T + (-0.142 - 0.989i)T^{2} \) |
| 47 | \( 1 + (0.0688 + 1.44i)T + (-0.995 + 0.0950i)T^{2} \) |
| 53 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 59 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 61 | \( 1 + (-0.223 + 0.175i)T + (0.235 - 0.971i)T^{2} \) |
| 71 | \( 1 + (-0.928 + 0.371i)T^{2} \) |
| 73 | \( 1 + (-0.235 + 0.971i)T^{2} \) |
| 79 | \( 1 + (-0.0475 + 0.998i)T^{2} \) |
| 83 | \( 1 + (1.07 - 0.431i)T + (0.723 - 0.690i)T^{2} \) |
| 89 | \( 1 + (-0.698 - 0.449i)T + (0.415 + 0.909i)T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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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
−10.34521238144288781344349321272, −9.702273095697595424005461388531, −8.420656539683219899652707320972, −7.16187440726744580019121960051, −6.28266358062449071308997293917, −5.75318202410758713028243877112, −5.07530233711153904371635775981, −4.31498670962769019569976179966, −3.33258446578764174777502529721, −2.01354892764832860615017010315,
1.08603067402916226977601877071, 1.85454962240754374564148259133, 4.05724072149076595423949179992, 4.67636973665114208126128206626, 5.33133143863428216000728304224, 6.11860717958150089025888823781, 7.03273363425177367925039507012, 7.67088646213464701249170317252, 8.156355115372262800812875333487, 10.06990712171357674737407555109