| L(s) = 1 | + 4·4-s − 9·5-s − 5·7-s + 9·9-s + 3·11-s + 16·16-s + 15·17-s − 19·19-s − 36·20-s − 30·23-s + 56·25-s − 20·28-s + 45·35-s + 36·36-s − 85·43-s + 12·44-s − 81·45-s + 75·47-s − 24·49-s − 27·55-s + 103·61-s − 45·63-s + 64·64-s + 60·68-s − 25·73-s − 76·76-s − 15·77-s + ⋯ |
| L(s) = 1 | + 4-s − 9/5·5-s − 5/7·7-s + 9-s + 3/11·11-s + 16-s + 0.882·17-s − 19-s − 9/5·20-s − 1.30·23-s + 2.23·25-s − 5/7·28-s + 9/7·35-s + 36-s − 1.97·43-s + 3/11·44-s − 9/5·45-s + 1.59·47-s − 0.489·49-s − 0.490·55-s + 1.68·61-s − 5/7·63-s + 64-s + 0.882·68-s − 0.342·73-s − 76-s − 0.194·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8253052350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8253052350\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + p T \) |
| good | 2 | \( ( 1 - p T )( 1 + p T ) \) |
| 3 | \( ( 1 - p T )( 1 + p T ) \) |
| 5 | \( 1 + 9 T + p^{2} T^{2} \) |
| 7 | \( 1 + 5 T + p^{2} T^{2} \) |
| 11 | \( 1 - 3 T + p^{2} T^{2} \) |
| 13 | \( ( 1 - p T )( 1 + p T ) \) |
| 17 | \( 1 - 15 T + p^{2} T^{2} \) |
| 23 | \( 1 + 30 T + p^{2} T^{2} \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( ( 1 - p T )( 1 + p T ) \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( 1 + 85 T + p^{2} T^{2} \) |
| 47 | \( 1 - 75 T + p^{2} T^{2} \) |
| 53 | \( ( 1 - p T )( 1 + p T ) \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 - 103 T + p^{2} T^{2} \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 + 25 T + p^{2} T^{2} \) |
| 79 | \( ( 1 - p T )( 1 + p T ) \) |
| 83 | \( 1 - 90 T + p^{2} T^{2} \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( ( 1 - p T )( 1 + p T ) \) |
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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
−18.82594982893777712386983113665, −16.53864180027817260860721905679, −15.83781470829667949819879005026, −14.93870523868892488078489132571, −12.59417096641023604631179220979, −11.74726308346753769195477735938, −10.29893100047884113540847229545, −7.971066154925277295460153217485, −6.77923630097008860108001780579, −3.78194741089294871802621036164,
3.78194741089294871802621036164, 6.77923630097008860108001780579, 7.971066154925277295460153217485, 10.29893100047884113540847229545, 11.74726308346753769195477735938, 12.59417096641023604631179220979, 14.93870523868892488078489132571, 15.83781470829667949819879005026, 16.53864180027817260860721905679, 18.82594982893777712386983113665
