| L(s) = 1 | + 3.23·5-s + 3.80·7-s + 1.45·11-s + 13-s − 2.47·17-s − 8.50·19-s + 5.47·25-s + 4·29-s + 3.80·31-s + 12.3·35-s + 6.94·37-s + 0.763·41-s + 4.70·43-s + 10.8·47-s + 7.47·49-s + 12.9·53-s + 4.70·55-s − 10.8·59-s − 4.47·61-s + 3.23·65-s − 6.71·67-s + 1.45·71-s + 6·73-s + 5.52·77-s − 9.40·79-s − 13.7·83-s − 8.00·85-s + ⋯ |
| L(s) = 1 | + 1.44·5-s + 1.43·7-s + 0.438·11-s + 0.277·13-s − 0.599·17-s − 1.95·19-s + 1.09·25-s + 0.742·29-s + 0.683·31-s + 2.08·35-s + 1.14·37-s + 0.119·41-s + 0.717·43-s + 1.58·47-s + 1.06·49-s + 1.77·53-s + 0.634·55-s − 1.41·59-s − 0.572·61-s + 0.401·65-s − 0.819·67-s + 0.172·71-s + 0.702·73-s + 0.629·77-s − 1.05·79-s − 1.51·83-s − 0.867·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.152927034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.152927034\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 3.23T + 5T^{2} \) |
| 7 | \( 1 - 3.80T + 7T^{2} \) |
| 11 | \( 1 - 1.45T + 11T^{2} \) |
| 17 | \( 1 + 2.47T + 17T^{2} \) |
| 19 | \( 1 + 8.50T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 - 3.80T + 31T^{2} \) |
| 37 | \( 1 - 6.94T + 37T^{2} \) |
| 41 | \( 1 - 0.763T + 41T^{2} \) |
| 43 | \( 1 - 4.70T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 4.47T + 61T^{2} \) |
| 67 | \( 1 + 6.71T + 67T^{2} \) |
| 71 | \( 1 - 1.45T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 9.40T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 - 2.29T + 89T^{2} \) |
| 97 | \( 1 + 6.94T + 97T^{2} \) |
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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
−8.700075396180834763097632024212, −7.898485433371543342913395616349, −6.91381637305858381038598942961, −6.14227281175591802589161496173, −5.69288337262947576911080203062, −4.57992593723754202085822872252, −4.25579633161248005433528843459, −2.59097392138320582759129205234, −2.02130609492664243517935354830, −1.12979216747129702888898773929,
1.12979216747129702888898773929, 2.02130609492664243517935354830, 2.59097392138320582759129205234, 4.25579633161248005433528843459, 4.57992593723754202085822872252, 5.69288337262947576911080203062, 6.14227281175591802589161496173, 6.91381637305858381038598942961, 7.898485433371543342913395616349, 8.700075396180834763097632024212
