| L(s) = 1 | − 3-s + 2·5-s + 7-s + 9-s − 6·11-s − 7·13-s − 2·15-s − 2·17-s − 21-s − 25-s − 27-s + 10·29-s − 8·31-s + 6·33-s + 2·35-s − 7·37-s + 7·39-s + 9·43-s + 2·45-s + 8·47-s − 6·49-s + 2·51-s − 2·53-s − 12·55-s − 4·59-s − 7·61-s + 63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.80·11-s − 1.94·13-s − 0.516·15-s − 0.485·17-s − 0.218·21-s − 1/5·25-s − 0.192·27-s + 1.85·29-s − 1.43·31-s + 1.04·33-s + 0.338·35-s − 1.15·37-s + 1.12·39-s + 1.37·43-s + 0.298·45-s + 1.16·47-s − 6/7·49-s + 0.280·51-s − 0.274·53-s − 1.61·55-s − 0.520·59-s − 0.896·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9976328745\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9976328745\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−16.25764244069753, −15.72966598050433, −15.31644749197281, −14.49342279515687, −14.04339448596368, −13.48276345571038, −12.70655509098671, −12.47971215086220, −11.82648375261780, −10.92569604525607, −10.54542361564673, −10.04849514967691, −9.518342642326526, −8.782622140824635, −7.923332911373477, −7.426265774165605, −6.876251174564990, −5.941057285923992, −5.435901561724697, −4.896370598122291, −4.440905681950506, −3.060477128280701, −2.382389765689260, −1.851738228592369, −0.4357867477575646,
0.4357867477575646, 1.851738228592369, 2.382389765689260, 3.060477128280701, 4.440905681950506, 4.896370598122291, 5.435901561724697, 5.941057285923992, 6.876251174564990, 7.426265774165605, 7.923332911373477, 8.782622140824635, 9.518342642326526, 10.04849514967691, 10.54542361564673, 10.92569604525607, 11.82648375261780, 12.47971215086220, 12.70655509098671, 13.48276345571038, 14.04339448596368, 14.49342279515687, 15.31644749197281, 15.72966598050433, 16.25764244069753
