| L(s) = 1 | + 5-s − 2·7-s + 2·13-s + 6·17-s + 6·23-s + 25-s + 4·29-s − 2·35-s + 10·37-s + 8·41-s − 2·43-s − 2·47-s − 3·49-s + 2·53-s + 14·61-s + 2·65-s − 4·67-s + 12·71-s + 6·73-s + 8·79-s + 2·83-s + 6·85-s + 8·89-s − 4·91-s − 6·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s + 0.554·13-s + 1.45·17-s + 1.25·23-s + 1/5·25-s + 0.742·29-s − 0.338·35-s + 1.64·37-s + 1.24·41-s − 0.304·43-s − 0.291·47-s − 3/7·49-s + 0.274·53-s + 1.79·61-s + 0.248·65-s − 0.488·67-s + 1.42·71-s + 0.702·73-s + 0.900·79-s + 0.219·83-s + 0.650·85-s + 0.847·89-s − 0.419·91-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.265170927\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.265170927\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 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
−12.84971559107469, −12.52567142912777, −11.89848315470130, −11.43213342077009, −10.91952771426192, −10.53115848630109, −9.911350676042824, −9.591843589114514, −9.289063565389056, −8.628973361454171, −8.115326146800221, −7.735144714146553, −7.050881288185529, −6.661711903905233, −6.100403296663957, −5.803610906359500, −5.117867693437806, −4.779877726264080, −3.907879398396265, −3.565431611169295, −2.891984208297409, −2.582357566222643, −1.745027252302753, −0.8968048695071398, −0.7423835940260190,
0.7423835940260190, 0.8968048695071398, 1.745027252302753, 2.582357566222643, 2.891984208297409, 3.565431611169295, 3.907879398396265, 4.779877726264080, 5.117867693437806, 5.803610906359500, 6.100403296663957, 6.661711903905233, 7.050881288185529, 7.735144714146553, 8.115326146800221, 8.628973361454171, 9.289063565389056, 9.591843589114514, 9.911350676042824, 10.53115848630109, 10.91952771426192, 11.43213342077009, 11.89848315470130, 12.52567142912777, 12.84971559107469
