| L(s) = 1 | + 7·7-s + 9·9-s − 6·11-s − 18·23-s + 25·25-s + 54·29-s + 38·37-s + 58·43-s + 49·49-s + 6·53-s + 63·63-s − 118·67-s − 114·71-s − 42·77-s + 94·79-s + 81·81-s − 54·99-s + 186·107-s − 106·109-s − 222·113-s + ⋯ |
| L(s) = 1 | + 7-s + 9-s − 0.545·11-s − 0.782·23-s + 25-s + 1.86·29-s + 1.02·37-s + 1.34·43-s + 49-s + 6/53·53-s + 63-s − 1.76·67-s − 1.60·71-s − 0.545·77-s + 1.18·79-s + 81-s − 0.545·99-s + 1.73·107-s − 0.972·109-s − 1.96·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.050947781\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.050947781\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 3 | \( ( 1 - p T )( 1 + p T ) \) |
| 5 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( 1 + 6 T + p^{2} T^{2} \) |
| 13 | \( ( 1 - p T )( 1 + p T ) \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( 1 + 18 T + p^{2} T^{2} \) |
| 29 | \( 1 - 54 T + p^{2} T^{2} \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( 1 - 38 T + p^{2} T^{2} \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( 1 - 58 T + p^{2} T^{2} \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( 1 - 6 T + p^{2} T^{2} \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( ( 1 - p T )( 1 + p T ) \) |
| 67 | \( 1 + 118 T + p^{2} T^{2} \) |
| 71 | \( 1 + 114 T + p^{2} T^{2} \) |
| 73 | \( ( 1 - p T )( 1 + p T ) \) |
| 79 | \( 1 - 94 T + p^{2} T^{2} \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 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
−10.72045145939386537215496425429, −10.20243305454671662989879543078, −9.031450746031776925013380373775, −8.058286713332828245914050735120, −7.34899742605760727974582150014, −6.20404979957876406574409948224, −4.94568834683337146243943156654, −4.22645053298129115296744546119, −2.58589390219691219925871406948, −1.18179655461210441946016712612,
1.18179655461210441946016712612, 2.58589390219691219925871406948, 4.22645053298129115296744546119, 4.94568834683337146243943156654, 6.20404979957876406574409948224, 7.34899742605760727974582150014, 8.058286713332828245914050735120, 9.031450746031776925013380373775, 10.20243305454671662989879543078, 10.72045145939386537215496425429
