| L(s) = 1 | − 3.37·5-s − 7-s + 4.37·13-s − 17-s + 4.74·19-s + 0.372·23-s + 6.37·25-s − 6.37·29-s − 8.37·31-s + 3.37·35-s − 1.37·37-s + 12.1·41-s + 3.74·43-s − 5.37·47-s + 49-s + 5.11·53-s − 11.7·59-s + 12.7·61-s − 14.7·65-s − 10.3·67-s − 1.62·71-s + 10·73-s − 8.11·79-s − 12.8·83-s + 3.37·85-s − 8.37·89-s − 4.37·91-s + ⋯ |
| L(s) = 1 | − 1.50·5-s − 0.377·7-s + 1.21·13-s − 0.242·17-s + 1.08·19-s + 0.0776·23-s + 1.27·25-s − 1.18·29-s − 1.50·31-s + 0.570·35-s − 0.225·37-s + 1.89·41-s + 0.571·43-s − 0.783·47-s + 0.142·49-s + 0.702·53-s − 1.52·59-s + 1.63·61-s − 1.82·65-s − 1.26·67-s − 0.193·71-s + 1.17·73-s − 0.913·79-s − 1.41·83-s + 0.365·85-s − 0.887·89-s − 0.458·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + 3.37T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4.37T + 13T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 19 | \( 1 - 4.74T + 19T^{2} \) |
| 23 | \( 1 - 0.372T + 23T^{2} \) |
| 29 | \( 1 + 6.37T + 29T^{2} \) |
| 31 | \( 1 + 8.37T + 31T^{2} \) |
| 37 | \( 1 + 1.37T + 37T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 43 | \( 1 - 3.74T + 43T^{2} \) |
| 47 | \( 1 + 5.37T + 47T^{2} \) |
| 53 | \( 1 - 5.11T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 + 1.62T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 8.11T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 + 8.37T + 89T^{2} \) |
| 97 | \( 1 + 13.4T + 97T^{2} \) |
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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
−8.295677556840947776220722027118, −7.52659687321100870708055044611, −7.11205962859574104311465217125, −6.04231452713929068613488621839, −5.30267062943731806230074644355, −4.07701585976052204432811833329, −3.75590218098522388230191643648, −2.83907479515440083565343748314, −1.29551271199322453616608660683, 0,
1.29551271199322453616608660683, 2.83907479515440083565343748314, 3.75590218098522388230191643648, 4.07701585976052204432811833329, 5.30267062943731806230074644355, 6.04231452713929068613488621839, 7.11205962859574104311465217125, 7.52659687321100870708055044611, 8.295677556840947776220722027118
