| L(s) = 1 | + 1.23·5-s − 7-s + 5.23·11-s − 0.763·13-s − 7.23·17-s + 19-s + 3.23·23-s − 3.47·25-s − 8.47·29-s + 0.472·31-s − 1.23·35-s + 8.94·37-s − 2·41-s − 4·43-s + 6.47·47-s + 49-s + 0.472·53-s + 6.47·55-s + 8·59-s + 8.47·61-s − 0.944·65-s + 0.763·67-s + 1.52·71-s + 4.47·73-s − 5.23·77-s + 15.7·79-s − 2·83-s + ⋯ |
| L(s) = 1 | + 0.552·5-s − 0.377·7-s + 1.57·11-s − 0.211·13-s − 1.75·17-s + 0.229·19-s + 0.674·23-s − 0.694·25-s − 1.57·29-s + 0.0847·31-s − 0.208·35-s + 1.47·37-s − 0.312·41-s − 0.609·43-s + 0.944·47-s + 0.142·49-s + 0.0648·53-s + 0.872·55-s + 1.04·59-s + 1.08·61-s − 0.117·65-s + 0.0933·67-s + 0.181·71-s + 0.523·73-s − 0.596·77-s + 1.76·79-s − 0.219·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.194295092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.194295092\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 1.23T + 5T^{2} \) |
| 11 | \( 1 - 5.23T + 11T^{2} \) |
| 13 | \( 1 + 0.763T + 13T^{2} \) |
| 17 | \( 1 + 7.23T + 17T^{2} \) |
| 23 | \( 1 - 3.23T + 23T^{2} \) |
| 29 | \( 1 + 8.47T + 29T^{2} \) |
| 31 | \( 1 - 0.472T + 31T^{2} \) |
| 37 | \( 1 - 8.94T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 6.47T + 47T^{2} \) |
| 53 | \( 1 - 0.472T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 - 8.47T + 61T^{2} \) |
| 67 | \( 1 - 0.763T + 67T^{2} \) |
| 71 | \( 1 - 1.52T + 71T^{2} \) |
| 73 | \( 1 - 4.47T + 73T^{2} \) |
| 79 | \( 1 - 15.7T + 79T^{2} \) |
| 83 | \( 1 + 2T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 - 1.23T + 97T^{2} \) |
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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
−7.55695829701518011535743341331, −6.84158340841337677529916553504, −6.42050509685313391210844701869, −5.76824538076009081581471509434, −4.92582765674797315601368394393, −4.08577905614069686340656597051, −3.59705079604356304886173667533, −2.42948670189715872435204141802, −1.84007508218933729871461112504, −0.70503064142347803198283574142,
0.70503064142347803198283574142, 1.84007508218933729871461112504, 2.42948670189715872435204141802, 3.59705079604356304886173667533, 4.08577905614069686340656597051, 4.92582765674797315601368394393, 5.76824538076009081581471509434, 6.42050509685313391210844701869, 6.84158340841337677529916553504, 7.55695829701518011535743341331
