| L(s) = 1 | − 2-s − 4-s − 7-s + 3·8-s + 11-s − 13-s + 14-s − 16-s − 5·19-s − 22-s + 23-s + 26-s + 28-s + 5·29-s − 2·31-s − 5·32-s + 4·37-s + 5·38-s + 5·41-s + 9·43-s − 44-s − 46-s − 6·47-s − 6·49-s + 52-s + 2·53-s − 3·56-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.377·7-s + 1.06·8-s + 0.301·11-s − 0.277·13-s + 0.267·14-s − 1/4·16-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 + 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 + 0.138·52-s + 0.274·53-s − 0.400·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 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
−8.004819412584731290661091407044, −7.29742373656822014066497252328, −6.49811005933712420621745068101, −5.79964654393675512717580365011, −4.71237175062719400684561071248, −4.28837541819658346201034530175, −3.27564888430751054574439464309, −2.22295382982141256983342070394, −1.12159600081611442837319704696, 0,
1.12159600081611442837319704696, 2.22295382982141256983342070394, 3.27564888430751054574439464309, 4.28837541819658346201034530175, 4.71237175062719400684561071248, 5.79964654393675512717580365011, 6.49811005933712420621745068101, 7.29742373656822014066497252328, 8.004819412584731290661091407044
