| L(s) = 1 | − 4-s − 2·9-s − 11-s − 7·13-s + 16-s + 17-s + 5·19-s + 25-s + 3·27-s − 31-s + 2·36-s + 16·37-s + 44-s + 47-s − 10·49-s + 7·52-s − 64-s − 9·67-s − 68-s − 21·71-s − 5·76-s + 4·81-s + 2·99-s − 100-s + 4·101-s − 3·108-s − 10·109-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 2/3·9-s − 0.301·11-s − 1.94·13-s + 1/4·16-s + 0.242·17-s + 1.14·19-s + 1/5·25-s + 0.577·27-s − 0.179·31-s + 1/3·36-s + 2.63·37-s + 0.150·44-s + 0.145·47-s − 1.42·49-s + 0.970·52-s − 1/8·64-s − 1.09·67-s − 0.121·68-s − 2.49·71-s − 0.573·76-s + 4/9·81-s + 0.201·99-s − 0.0999·100-s + 0.398·101-s − 0.288·108-s − 0.957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 295788 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 295788 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 157 | $C_2$ | \( 1 - 13 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 85 T^{2} + p^{2} T^{4} \) |
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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
−8.642331927328580378123176055660, −8.035717480534527668775262437968, −7.66893614013298027561709536746, −7.39855192167583514711505536731, −6.78147664391973395610477166307, −6.09894343146060574501405601442, −5.68789335797218676311795550418, −5.13744500267665327495241770421, −4.68226235857511659628650928104, −4.33326523196144528452610683472, −3.34844152208132820450281506736, −2.85277063283961356928576782534, −2.40396486627753918954513809857, −1.19965964979363041173447755758, 0,
1.19965964979363041173447755758, 2.40396486627753918954513809857, 2.85277063283961356928576782534, 3.34844152208132820450281506736, 4.33326523196144528452610683472, 4.68226235857511659628650928104, 5.13744500267665327495241770421, 5.68789335797218676311795550418, 6.09894343146060574501405601442, 6.78147664391973395610477166307, 7.39855192167583514711505536731, 7.66893614013298027561709536746, 8.035717480534527668775262437968, 8.642331927328580378123176055660
