L(s) = 1 | + (1.05 + 0.946i)2-s + (0.207 + 1.98i)4-s + 1.60i·5-s + (−1.66 + 2.28i)8-s + (−1.52 + 1.68i)10-s − 2.67i·11-s + 3.37i·13-s + (−3.91 + 0.823i)16-s + 1.60i·17-s − 4.30·19-s + (−3.19 + 0.333i)20-s + (2.53 − 2.81i)22-s + 6.46i·23-s + 2.41·25-s + (−3.19 + 3.54i)26-s + ⋯ |
L(s) = 1 | + (0.742 + 0.669i)2-s + (0.103 + 0.994i)4-s + 0.719i·5-s + (−0.588 + 0.808i)8-s + (−0.481 + 0.534i)10-s − 0.807i·11-s + 0.937i·13-s + (−0.978 + 0.205i)16-s + 0.390i·17-s − 0.987·19-s + (−0.715 + 0.0744i)20-s + (0.540 − 0.599i)22-s + 1.34i·23-s + 0.482·25-s + (−0.627 + 0.696i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0534i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0534i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.881195657\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.881195657\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.05 - 0.946i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 1.60iT - 5T^{2} \) |
| 11 | \( 1 + 2.67iT - 11T^{2} \) |
| 13 | \( 1 - 3.37iT - 13T^{2} \) |
| 17 | \( 1 - 1.60iT - 17T^{2} \) |
| 19 | \( 1 + 4.30T + 19T^{2} \) |
| 23 | \( 1 - 6.46iT - 23T^{2} \) |
| 29 | \( 1 - 4.71T + 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 - 0.242T + 37T^{2} \) |
| 41 | \( 1 - 11.6iT - 41T^{2} \) |
| 43 | \( 1 + 7.95iT - 43T^{2} \) |
| 47 | \( 1 + 9.04T + 47T^{2} \) |
| 53 | \( 1 + 2.46T + 53T^{2} \) |
| 59 | \( 1 - 9.04T + 59T^{2} \) |
| 61 | \( 1 + 3.37iT - 61T^{2} \) |
| 67 | \( 1 - 11.2iT - 67T^{2} \) |
| 71 | \( 1 + 8.03iT - 71T^{2} \) |
| 73 | \( 1 + 6.17iT - 73T^{2} \) |
| 79 | \( 1 - 3.29iT - 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 - 0.275iT - 89T^{2} \) |
| 97 | \( 1 + 12.9iT - 97T^{2} \) |
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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
−9.501769116568521949988464324433, −8.699925381516528639526706895661, −7.995095276925466720412160948325, −7.04919399465524295611603993182, −6.51938054863290119959736528671, −5.77223801054791571313272264896, −4.82979509922893565702613771704, −3.81673782535829819009238969241, −3.15956853086506842858357984422, −1.95860187502535794551826074907,
0.51609221502480383096396745285, 1.85529435415175420698527321478, 2.82256756250852553413518406581, 3.96152323348745887392230972105, 4.76942666096787570465839487927, 5.33609522187350328319099596813, 6.35926894887145491067131184698, 7.14804926253507554106939902133, 8.304211172496932192925059732920, 9.012743990335537815443476512045