L(s) = 1 | + (−0.809 − 0.587i)2-s + (−0.5 − 1.53i)3-s + (0.309 + 0.951i)4-s + (−0.5 + 1.53i)6-s + 0.618·7-s + (0.309 − 0.951i)8-s + (−1.30 + 0.951i)9-s + (1.30 − 0.951i)12-s + (−0.500 − 0.363i)14-s + (−0.809 + 0.587i)16-s + 1.61·18-s + (−0.309 − 0.951i)21-s + (−0.5 − 0.363i)23-s − 1.61·24-s + (0.809 + 0.587i)27-s + (0.190 + 0.587i)28-s + ⋯ |
L(s) = 1 | + (−0.809 − 0.587i)2-s + (−0.5 − 1.53i)3-s + (0.309 + 0.951i)4-s + (−0.5 + 1.53i)6-s + 0.618·7-s + (0.309 − 0.951i)8-s + (−1.30 + 0.951i)9-s + (1.30 − 0.951i)12-s + (−0.500 − 0.363i)14-s + (−0.809 + 0.587i)16-s + 1.61·18-s + (−0.309 − 0.951i)21-s + (−0.5 − 0.363i)23-s − 1.61·24-s + (0.809 + 0.587i)27-s + (0.190 + 0.587i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5495875884\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5495875884\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 - 0.618T + T^{2} \) |
| 11 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 29 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 + 1.61T + T^{2} \) |
| 47 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)T^{2} \) |
| 71 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 97 | \( 1 + (0.809 - 0.587i)T^{2} \) |
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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
−8.439383143053958352018869546685, −7.970284321699812641444748244687, −7.37190564555950334478913277367, −6.60006281714236366141676267425, −5.91519100068082864209217899615, −4.76434976190238893648256653769, −3.57870559537665670100833798410, −2.27050737583388664364618984720, −1.76415357208800149807363057969, −0.51602764847268626932077886377,
1.51541286343495239524025177349, 3.03417081999184028604173930513, 4.21342182419586493320455145483, 4.93527975255815716123480287763, 5.53751144888620573910112577479, 6.31542385665047452944571800113, 7.30238035993669576693396787361, 8.167646138181221618144939616928, 8.829117830064660071121801290564, 9.621630258559279268550307350265