| L(s) = 1 | − 3-s − 3.61·5-s − 7-s + 9-s + 11-s + 5.09·13-s + 3.61·15-s − 5·17-s + 3.47·19-s + 21-s − 23-s + 8.09·25-s − 27-s − 6.23·29-s + 8.70·31-s − 33-s + 3.61·35-s + 1.47·37-s − 5.09·39-s − 5.76·41-s − 6.32·43-s − 3.61·45-s − 3.70·47-s + 49-s + 5·51-s + 0.381·53-s − 3.61·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.61·5-s − 0.377·7-s + 0.333·9-s + 0.301·11-s + 1.41·13-s + 0.934·15-s − 1.21·17-s + 0.796·19-s + 0.218·21-s − 0.208·23-s + 1.61·25-s − 0.192·27-s − 1.15·29-s + 1.56·31-s − 0.174·33-s + 0.611·35-s + 0.242·37-s − 0.815·39-s − 0.900·41-s − 0.964·43-s − 0.539·45-s − 0.540·47-s + 0.142·49-s + 0.700·51-s + 0.0524·53-s − 0.487·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7978637677\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7978637677\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 3.61T + 5T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 - 5.09T + 13T^{2} \) |
| 17 | \( 1 + 5T + 17T^{2} \) |
| 19 | \( 1 - 3.47T + 19T^{2} \) |
| 29 | \( 1 + 6.23T + 29T^{2} \) |
| 31 | \( 1 - 8.70T + 31T^{2} \) |
| 37 | \( 1 - 1.47T + 37T^{2} \) |
| 41 | \( 1 + 5.76T + 41T^{2} \) |
| 43 | \( 1 + 6.32T + 43T^{2} \) |
| 47 | \( 1 + 3.70T + 47T^{2} \) |
| 53 | \( 1 - 0.381T + 53T^{2} \) |
| 59 | \( 1 - 3.61T + 59T^{2} \) |
| 61 | \( 1 + 7.56T + 61T^{2} \) |
| 67 | \( 1 - 7.32T + 67T^{2} \) |
| 71 | \( 1 - 4.32T + 71T^{2} \) |
| 73 | \( 1 + 0.527T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 + 9.18T + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−8.015672636755053911095907203718, −6.95115601489987768791253286869, −6.69017447137998912765507837661, −5.84920794602332160283740966383, −4.94989970894876461359380500093, −4.16797382316433369939937273686, −3.72111112863357672990138547861, −2.94830031917551792198713365863, −1.52090931534457295814492378661, −0.47940065137436031845109445852,
0.47940065137436031845109445852, 1.52090931534457295814492378661, 2.94830031917551792198713365863, 3.72111112863357672990138547861, 4.16797382316433369939937273686, 4.94989970894876461359380500093, 5.84920794602332160283740966383, 6.69017447137998912765507837661, 6.95115601489987768791253286869, 8.015672636755053911095907203718
