| L(s) = 1 | − 4-s − 2·5-s − 4·11-s + 16-s − 12·19-s + 2·20-s − 25-s − 18·29-s − 4·31-s − 22·41-s + 4·44-s + 13·49-s + 8·55-s + 8·59-s − 14·61-s − 64-s − 12·71-s + 12·76-s + 24·79-s − 2·80-s − 2·89-s + 24·95-s + 100-s + 4·101-s − 14·109-s + 18·116-s − 10·121-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.894·5-s − 1.20·11-s + 1/4·16-s − 2.75·19-s + 0.447·20-s − 1/5·25-s − 3.34·29-s − 0.718·31-s − 3.43·41-s + 0.603·44-s + 13/7·49-s + 1.07·55-s + 1.04·59-s − 1.79·61-s − 1/8·64-s − 1.42·71-s + 1.37·76-s + 2.70·79-s − 0.223·80-s − 0.211·89-s + 2.46·95-s + 1/10·100-s + 0.398·101-s − 1.34·109-s + 1.67·116-s − 0.909·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−10.17883907488857588694831212536, −9.586506805980223172737277751495, −8.927895939250911386391102622991, −8.924112324095231111501790289339, −8.193778223586007174045619339017, −8.090914453733625316899062029754, −7.38424231783881445130406252641, −7.28747656041900917098478454085, −6.54201123385877517129664791782, −6.10521963207707746450126980464, −5.35628086966803970879463285667, −5.29633122825214424114841254839, −4.52829153612623436996189583037, −3.95206760983480762547768841836, −3.78168104628012741309522551527, −3.10427971779357953583827368712, −2.11786822605608749364722969462, −1.86627265350834450046512039765, 0, 0,
1.86627265350834450046512039765, 2.11786822605608749364722969462, 3.10427971779357953583827368712, 3.78168104628012741309522551527, 3.95206760983480762547768841836, 4.52829153612623436996189583037, 5.29633122825214424114841254839, 5.35628086966803970879463285667, 6.10521963207707746450126980464, 6.54201123385877517129664791782, 7.28747656041900917098478454085, 7.38424231783881445130406252641, 8.090914453733625316899062029754, 8.193778223586007174045619339017, 8.924112324095231111501790289339, 8.927895939250911386391102622991, 9.586506805980223172737277751495, 10.17883907488857588694831212536
