| L(s) = 1 | + 4·4-s + 5·7-s + 9·9-s + 3·11-s + 16·16-s − 15·17-s − 19·19-s + 30·23-s + 20·28-s + 36·36-s + 85·43-s + 12·44-s − 75·47-s − 24·49-s + 103·61-s + 45·63-s + 64·64-s − 60·68-s + 25·73-s − 76·76-s + 15·77-s + 81·81-s − 90·83-s + 120·92-s + 27·99-s − 102·101-s + 80·112-s + ⋯ |
| L(s) = 1 | + 4-s + 5/7·7-s + 9-s + 3/11·11-s + 16-s − 0.882·17-s − 19-s + 1.30·23-s + 5/7·28-s + 36-s + 1.97·43-s + 3/11·44-s − 1.59·47-s − 0.489·49-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 + 81-s − 1.08·83-s + 1.30·92-s + 3/11·99-s − 1.00·101-s + 5/7·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.576967847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.576967847\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 + p T \) |
| good | 2 | \( ( 1 - p T )( 1 + p T ) \) |
| 3 | \( ( 1 - p T )( 1 + p T ) \) |
| 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
−10.97824835758438456141092760639, −10.08927667814574676508112621268, −8.970942058832604939087759660821, −7.949838331132170786693754189568, −7.03533846751643783409921979374, −6.38863232915481744021472700460, −5.03408173624368701490905456831, −3.98661370310743534215797924373, −2.46204029384499909558709512759, −1.36505232125308717618727797155,
1.36505232125308717618727797155, 2.46204029384499909558709512759, 3.98661370310743534215797924373, 5.03408173624368701490905456831, 6.38863232915481744021472700460, 7.03533846751643783409921979374, 7.949838331132170786693754189568, 8.970942058832604939087759660821, 10.08927667814574676508112621268, 10.97824835758438456141092760639
