L(s) = 1 | + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)4-s + (−0.309 − 0.951i)5-s + (−1.30 − 0.951i)7-s + (0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + 0.999·10-s + (−0.809 − 0.587i)11-s + (−0.190 + 0.587i)13-s + (1.30 − 0.951i)14-s + (0.309 + 0.951i)16-s + (0.809 + 0.587i)18-s + (−0.5 + 0.363i)19-s + (−0.309 + 0.951i)20-s + (0.809 − 0.587i)22-s + 1.61·23-s + ⋯ |
L(s) = 1 | + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)4-s + (−0.309 − 0.951i)5-s + (−1.30 − 0.951i)7-s + (0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + 0.999·10-s + (−0.809 − 0.587i)11-s + (−0.190 + 0.587i)13-s + (1.30 − 0.951i)14-s + (0.309 + 0.951i)16-s + (0.809 + 0.587i)18-s + (−0.5 + 0.363i)19-s + (−0.309 + 0.951i)20-s + (0.809 − 0.587i)22-s + 1.61·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4888507206\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4888507206\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 - 0.951i)T \) |
| 5 | \( 1 + (0.309 + 0.951i)T \) |
| 11 | \( 1 + (0.809 + 0.587i)T \) |
good | 3 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 - 1.61T + T^{2} \) |
| 29 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 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 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 + 1.61T + T^{2} \) |
| 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
−10.99134793300760340446897719513, −9.966876871958335825954467510446, −9.303386987701724474840968131258, −8.542861214962785073772363154239, −7.39391248423991224895058578754, −6.69879066680353266736022461842, −5.70073743077930168777627439258, −4.47042225572496070240827824099, −3.55323119105194260276702961453, −0.72581805044540749428764785656,
2.53404863415168737796626306873, 2.93465299698354806766496049907, 4.47407103360813740605519599871, 5.69948686061222677001765067250, 7.08837833485904893352181719818, 7.85765543716049582894882216597, 9.034372668066997282817264643011, 9.873892412393368350242783401252, 10.61178511566919080633072644514, 11.21215458012189104285884269078