L(s) = 1 | + (−0.535 − 1.64i)2-s + (0.809 + 0.587i)3-s + (−1.61 + 1.17i)4-s + (0.309 − 0.951i)5-s + (0.535 − 1.64i)6-s + (1.40 + 1.01i)8-s + (0.309 + 0.951i)9-s − 1.73·10-s − 2·12-s + (0.809 − 0.587i)15-s + (0.309 − 0.951i)16-s + (0.535 − 1.64i)17-s + (1.40 − 1.01i)18-s + (0.618 + 1.90i)20-s + 23-s + (0.535 + 1.64i)24-s + ⋯ |
L(s) = 1 | + (−0.535 − 1.64i)2-s + (0.809 + 0.587i)3-s + (−1.61 + 1.17i)4-s + (0.309 − 0.951i)5-s + (0.535 − 1.64i)6-s + (1.40 + 1.01i)8-s + (0.309 + 0.951i)9-s − 1.73·10-s − 2·12-s + (0.809 − 0.587i)15-s + (0.309 − 0.951i)16-s + (0.535 − 1.64i)17-s + (1.40 − 1.01i)18-s + (0.618 + 1.90i)20-s + 23-s + (0.535 + 1.64i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.598 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.598 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.102509743\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.102509743\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.809 - 0.587i)T \) |
| 5 | \( 1 + (-0.309 + 0.951i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.535 + 1.64i)T + (-0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.535 + 1.64i)T + (-0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (0.535 - 1.64i)T + (-0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (-0.535 - 1.64i)T + (-0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 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
−9.458934793428176569096665162230, −8.797070615553972445640951090642, −8.119653333443037578986864693181, −7.21754372097546556185779578272, −5.47732914535586511191837208731, −4.72080104376627173785505137978, −3.93143525383236692756425740801, −2.96293786594915210712693435736, −2.22274269721346840641252272072, −1.02360391337064862099277586861,
1.52912091948771416664482748078, 2.92753295424197435362091785401, 3.94551032024463756987679604864, 5.29497581491717315039404967932, 6.22777166053227087526937226787, 6.60432712213299100013485298145, 7.49103474454405743743242412917, 7.87894482300309153960823155124, 8.796530974280093281531751669400, 9.282606015185497264360921776747