L(s) = 1 | + (0.809 − 0.587i)2-s + (0.5 + 0.866i)5-s + (−0.669 − 0.743i)7-s + (0.309 + 0.951i)8-s + (0.669 − 0.743i)9-s + (0.913 + 0.406i)10-s + (−0.978 − 0.207i)14-s + (0.809 + 0.587i)16-s + (0.104 − 0.994i)18-s + (0.104 + 0.994i)19-s + (0.309 − 0.951i)35-s + (0.669 + 0.743i)38-s + (−0.669 + 0.743i)40-s + (−0.913 − 0.406i)41-s + (0.978 + 0.207i)45-s + ⋯ |
L(s) = 1 | + (0.809 − 0.587i)2-s + (0.5 + 0.866i)5-s + (−0.669 − 0.743i)7-s + (0.309 + 0.951i)8-s + (0.669 − 0.743i)9-s + (0.913 + 0.406i)10-s + (−0.978 − 0.207i)14-s + (0.809 + 0.587i)16-s + (0.104 − 0.994i)18-s + (0.104 + 0.994i)19-s + (0.309 − 0.951i)35-s + (0.669 + 0.743i)38-s + (−0.669 + 0.743i)40-s + (−0.913 − 0.406i)41-s + (0.978 + 0.207i)45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.565690949\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.565690949\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 31 | \( 1 \) |
good | 2 | \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \) |
| 3 | \( 1 + (-0.669 + 0.743i)T^{2} \) |
| 5 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (0.669 + 0.743i)T + (-0.104 + 0.994i)T^{2} \) |
| 11 | \( 1 + (-0.913 + 0.406i)T^{2} \) |
| 13 | \( 1 + (0.978 + 0.207i)T^{2} \) |
| 17 | \( 1 + (-0.913 - 0.406i)T^{2} \) |
| 19 | \( 1 + (-0.104 - 0.994i)T + (-0.978 + 0.207i)T^{2} \) |
| 23 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.913 + 0.406i)T + (0.669 + 0.743i)T^{2} \) |
| 43 | \( 1 + (0.978 - 0.207i)T^{2} \) |
| 47 | \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (0.104 + 0.994i)T^{2} \) |
| 59 | \( 1 + (0.913 - 0.406i)T + (0.669 - 0.743i)T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.669 - 0.743i)T + (-0.104 - 0.994i)T^{2} \) |
| 73 | \( 1 + (-0.913 + 0.406i)T^{2} \) |
| 79 | \( 1 + (-0.913 - 0.406i)T^{2} \) |
| 83 | \( 1 + (-0.669 - 0.743i)T^{2} \) |
| 89 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
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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
−10.22520036209694947744195062403, −9.807526051751717952296328020812, −8.576769874602581919412023267016, −7.47277656510777758387005073915, −6.70843601474910535442376309376, −5.92609107125770357041190415887, −4.67262956409460185881218287176, −3.66391454847259638094659734680, −3.17670698977402126135125818953, −1.82322750323479001888303308449,
1.54376595017761818053877068744, 3.03803085588313957998326853375, 4.47662477328812333837678216858, 5.01593406507155790467970971622, 5.81994200762470407767016058452, 6.61160531332545103609261591002, 7.50874444286084597827315918927, 8.666034385778887218707763119738, 9.470019250949657410065883717322, 9.999292838019451830328868149768