L(s) = 1 | + (−1.07 + 0.623i)2-s + (0.277 − 0.480i)4-s + (−0.866 − 0.5i)7-s − 0.554i·8-s + (−0.5 + 0.866i)9-s + (−0.385 + 0.222i)11-s + 1.24·14-s + (0.623 + 1.07i)16-s − 1.24i·18-s + (0.277 − 0.480i)22-s + (−0.900 − 1.56i)23-s − 25-s + (−0.480 + 0.277i)28-s + (−0.623 − 1.07i)29-s + (−0.866 − 0.500i)32-s + ⋯ |
L(s) = 1 | + (−1.07 + 0.623i)2-s + (0.277 − 0.480i)4-s + (−0.866 − 0.5i)7-s − 0.554i·8-s + (−0.5 + 0.866i)9-s + (−0.385 + 0.222i)11-s + 1.24·14-s + (0.623 + 1.07i)16-s − 1.24i·18-s + (0.277 − 0.480i)22-s + (−0.900 − 1.56i)23-s − 25-s + (−0.480 + 0.277i)28-s + (−0.623 − 1.07i)29-s + (−0.866 − 0.500i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.08367349745\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.08367349745\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.866 + 0.5i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (1.07 - 0.623i)T + (0.5 - 0.866i)T^{2} \) |
| 3 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + T^{2} \) |
| 11 | \( 1 + (0.385 - 0.222i)T + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.900 + 1.56i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (1.56 - 0.900i)T + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.623 + 1.07i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + 1.80T + T^{2} \) |
| 59 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-1.56 + 0.900i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (1.56 + 0.900i)T + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + 0.445T + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
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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
−9.711812020526006979000810327422, −8.765608831128056468011028369169, −8.040470758506595155701475052002, −7.46984382745816095153772812332, −6.55057356344604010028470829213, −5.84628755419726075911258865361, −4.54067656031988442557029525854, −3.48235677859109244194749211733, −2.12859959644883054663443318055, −0.099738193046379258498698453951,
1.68635329227123306944097875268, 2.91110653708313057372189274667, 3.70786929091803621800811469312, 5.43751721768950781468816140397, 5.92945203360945626997423483691, 7.12640575960103850587468187061, 8.058901751796472869728486958359, 8.875817890168211659299209883873, 9.477291430814399749751881996181, 9.940097183891417691129218610769