| L(s) = 1 | + 2·2-s + 3-s + 2·6-s − 2·7-s − 4·8-s + 9-s − 4·14-s − 4·16-s + 2·18-s + 19-s − 2·21-s − 4·24-s + 9·25-s + 27-s + 10·29-s + 2·38-s + 4·41-s − 4·42-s + 2·43-s − 4·48-s − 7·49-s + 18·50-s + 6·53-s + 2·54-s + 8·56-s + 57-s + 20·58-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 0.816·6-s − 0.755·7-s − 1.41·8-s + 1/3·9-s − 1.06·14-s − 16-s + 0.471·18-s + 0.229·19-s − 0.436·21-s − 0.816·24-s + 9/5·25-s + 0.192·27-s + 1.85·29-s + 0.324·38-s + 0.624·41-s − 0.617·42-s + 0.304·43-s − 0.577·48-s − 49-s + 2.54·50-s + 0.824·53-s + 0.272·54-s + 1.06·56-s + 0.132·57-s + 2.62·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185193 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185193 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.054670288\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.054670288\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( 1 - T \) |
| 19 | $C_1$ | \( 1 - T \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 81 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
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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
−9.099556008782784866287192831342, −8.698848699364751345434557049907, −8.322612337412190178377081216167, −7.68021689605794645181036076605, −7.01404483688291530794040415586, −6.60272106343856494082476987480, −6.14335445915478683026868588811, −5.49392019345695010405887545895, −4.96803070911136097666889189539, −4.55105626500682201261421216238, −4.08306785775273967150431583812, −3.31158669960494150135264534484, −3.09219103599700765101313178030, −2.35752926810146580833085941312, −0.919755367728835716958271297122,
0.919755367728835716958271297122, 2.35752926810146580833085941312, 3.09219103599700765101313178030, 3.31158669960494150135264534484, 4.08306785775273967150431583812, 4.55105626500682201261421216238, 4.96803070911136097666889189539, 5.49392019345695010405887545895, 6.14335445915478683026868588811, 6.60272106343856494082476987480, 7.01404483688291530794040415586, 7.68021689605794645181036076605, 8.322612337412190178377081216167, 8.698848699364751345434557049907, 9.099556008782784866287192831342
