L(s) = 1 | + (−1.30 − 0.951i)2-s + (0.500 + 1.53i)4-s + (−0.309 − 0.951i)7-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (−0.499 + 1.53i)14-s + (0.5 + 1.53i)18-s − 1.61·23-s + (0.309 − 0.951i)25-s + (1.30 − 0.951i)28-s + (−0.190 − 0.587i)29-s − 0.999·32-s + (0.499 − 1.53i)36-s + (−0.5 − 1.53i)37-s − 0.618·43-s + ⋯ |
L(s) = 1 | + (−1.30 − 0.951i)2-s + (0.500 + 1.53i)4-s + (−0.309 − 0.951i)7-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (−0.499 + 1.53i)14-s + (0.5 + 1.53i)18-s − 1.61·23-s + (0.309 − 0.951i)25-s + (1.30 − 0.951i)28-s + (−0.190 − 0.587i)29-s − 0.999·32-s + (0.499 − 1.53i)36-s + (−0.5 − 1.53i)37-s − 0.618·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3091916357\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3091916357\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.309 + 0.951i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 3 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 5 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + 1.61T + T^{2} \) |
| 29 | \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 + 0.618T + T^{2} \) |
| 47 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 - 0.618T + T^{2} \) |
| 71 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
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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
−9.999918916586626457518071557147, −9.347189786269145117859156957906, −8.441270364421093327515619238717, −7.81756205265229096339741226985, −6.78344233225989769102366057380, −5.75962347876640331536145448310, −4.13143090436035810498936726619, −3.22508619898816023538158437224, −2.04569229335586695001424957120, −0.45754450192027808659234897582,
1.87488074131748201667848419479, 3.26398044503351980067398821437, 5.07012903673410262345504589415, 5.87556942901334309504114122056, 6.57562388831626833846385822526, 7.64397418199673817283771218072, 8.342472638051968131771842325024, 8.915279016991001173459354428843, 9.712694596399048929058284963843, 10.45590093473702290302599347479