| L(s) = 1 | + 2-s − 4-s − 7-s − 3·8-s − 3·9-s − 11-s − 13-s − 14-s − 16-s − 3·18-s − 5·19-s − 22-s − 23-s − 26-s + 28-s − 5·29-s − 2·31-s + 5·32-s + 3·36-s + 4·37-s − 5·38-s − 5·41-s + 9·43-s + 44-s − 46-s + 6·47-s − 6·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.377·7-s − 1.06·8-s − 9-s − 0.301·11-s − 0.277·13-s − 0.267·14-s − 1/4·16-s − 0.707·18-s − 1.14·19-s − 0.213·22-s − 0.208·23-s − 0.196·26-s + 0.188·28-s − 0.928·29-s − 0.359·31-s + 0.883·32-s + 1/2·36-s + 0.657·37-s − 0.811·38-s − 0.780·41-s + 1.37·43-s + 0.150·44-s − 0.147·46-s + 0.875·47-s − 6/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−10.33150215549316429282479405355, −9.275522442220237927748460955767, −8.670602879067891765462746149527, −7.63796816663650653601242372317, −6.28621747108681454608291048309, −5.64400527552337470878907609706, −4.61492894085925041951760672838, −3.58600778086620467377109651751, −2.51027297613032043342122660971, 0,
2.51027297613032043342122660971, 3.58600778086620467377109651751, 4.61492894085925041951760672838, 5.64400527552337470878907609706, 6.28621747108681454608291048309, 7.63796816663650653601242372317, 8.670602879067891765462746149527, 9.275522442220237927748460955767, 10.33150215549316429282479405355
