| L(s) = 1 | − 4-s + 5·9-s + 16-s − 2·19-s + 6·29-s + 4·31-s − 5·36-s + 12·41-s + 13·49-s − 6·59-s + 16·61-s − 64-s + 24·71-s + 2·76-s − 28·79-s + 16·81-s − 12·89-s + 24·101-s − 22·109-s − 6·116-s − 22·121-s − 4·124-s + 127-s + 131-s + 137-s + 139-s + 5·144-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 5/3·9-s + 1/4·16-s − 0.458·19-s + 1.11·29-s + 0.718·31-s − 5/6·36-s + 1.87·41-s + 13/7·49-s − 0.781·59-s + 2.04·61-s − 1/8·64-s + 2.84·71-s + 0.229·76-s − 3.15·79-s + 16/9·81-s − 1.27·89-s + 2.38·101-s − 2.10·109-s − 0.557·116-s − 2·121-s − 0.359·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5/12·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.248811955\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.248811955\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.14412570831628558915944559515, −9.871140753453987620504090159995, −9.499938893504928702570356864974, −9.022994393690980526882549759416, −8.632754664683448521621364142105, −8.112494231222844401996753182294, −7.80253861516502004887216369859, −7.27299756019718214529961900449, −6.79808793499429898924222457416, −6.62009187741297926155647777340, −5.84379151744724813970441112931, −5.55267944731353413432100877937, −4.71600772933333300382195326786, −4.60404974447699729840986976376, −3.90831902467415936018583668795, −3.80258792603032100861754452958, −2.71880443125033685463285670275, −2.34385773632888722896866027426, −1.37013021504630416699966885410, −0.803118169931028016795268971038,
0.803118169931028016795268971038, 1.37013021504630416699966885410, 2.34385773632888722896866027426, 2.71880443125033685463285670275, 3.80258792603032100861754452958, 3.90831902467415936018583668795, 4.60404974447699729840986976376, 4.71600772933333300382195326786, 5.55267944731353413432100877937, 5.84379151744724813970441112931, 6.62009187741297926155647777340, 6.79808793499429898924222457416, 7.27299756019718214529961900449, 7.80253861516502004887216369859, 8.112494231222844401996753182294, 8.632754664683448521621364142105, 9.022994393690980526882549759416, 9.499938893504928702570356864974, 9.871140753453987620504090159995, 10.14412570831628558915944559515
