L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)7-s − 0.999·8-s + (−0.866 − 0.5i)9-s + (−0.366 + 1.36i)11-s + (−1 − i)13-s + (0.499 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.866 + 0.499i)18-s + (0.366 + 1.36i)19-s + (0.999 + i)22-s + (−0.5 + 0.866i)23-s + (−1.36 + 0.366i)26-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)7-s − 0.999·8-s + (−0.866 − 0.5i)9-s + (−0.366 + 1.36i)11-s + (−1 − i)13-s + (0.499 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.866 + 0.499i)18-s + (0.366 + 1.36i)19-s + (0.999 + i)22-s + (−0.5 + 0.866i)23-s + (−1.36 + 0.366i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6749087955\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6749087955\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
good | 3 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 13 | \( 1 + (1 + i)T + iT^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + iT^{2} \) |
| 31 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 - iT - T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 59 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 67 | \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (1 - i)T - iT^{2} \) |
| 89 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−9.495471507388190516460277237331, −8.373807706678552571696192013294, −7.79738944661258690949505407303, −6.55376239455356186678715473213, −5.67308156915301121505459167734, −5.39579716655844131179020963547, −4.27114579421168370294612389037, −3.27809673364695660924127950647, −2.65981476218263716469426807442, −1.65628115647667213362865988380,
0.34183864428953492738937734453, 2.69700534836833561565620804690, 3.16170202199803312175458061135, 4.38535075589039071647239101963, 4.98443025947301964755874867121, 5.82398691194308524621286555568, 6.63585852537412867812935293153, 7.24301521452979690435384804692, 7.962783791500464553259607524421, 8.744642548854449339674500044566