L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.965 − 1.67i)3-s + (−0.499 − 0.866i)4-s + (0.866 − 1.5i)5-s − 1.93·6-s + (0.258 + 0.965i)7-s − 0.999·8-s + (−1.36 + 2.36i)9-s + (−0.866 − 1.5i)10-s + (−0.258 − 0.448i)11-s + (−0.965 + 1.67i)12-s + (0.965 + 0.258i)14-s − 3.34·15-s + (−0.5 + 0.866i)16-s + (1.36 + 2.36i)18-s + (0.258 − 0.448i)19-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.965 − 1.67i)3-s + (−0.499 − 0.866i)4-s + (0.866 − 1.5i)5-s − 1.93·6-s + (0.258 + 0.965i)7-s − 0.999·8-s + (−1.36 + 2.36i)9-s + (−0.866 − 1.5i)10-s + (−0.258 − 0.448i)11-s + (−0.965 + 1.67i)12-s + (0.965 + 0.258i)14-s − 3.34·15-s + (−0.5 + 0.866i)16-s + (1.36 + 2.36i)18-s + (0.258 − 0.448i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.750 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.750 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9728526455\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9728526455\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.258 - 0.965i)T \) |
| 41 | \( 1 - T \) |
good | 3 | \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - 1.93T + T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.440653766532190890426792394619, −8.714440068813316676325374459663, −8.020282238570439747623032287164, −6.63272392355241581661089153855, −5.65129426239334469057065288301, −5.53228506996965252126231164317, −4.66694523841506889504044113411, −2.58665123259185702685698889829, −1.81937357046570986560164506648, −0.866174618577696087774110532668,
2.91138898923709232084090716102, 3.79541892039467770517004050140, 4.56650684387378521796771055308, 5.49748251868575040280317254808, 6.15951033267591336138433258016, 6.86619554014286051767549694693, 7.75753123428272403574264355514, 9.198203905077416404482021584055, 9.829715363034550421037420676803, 10.47094033174420814752691764867