| L(s) = 1 | − 2-s + 2·5-s + 8-s − 9-s − 2·10-s + 13-s − 16-s − 17-s + 18-s + 25-s − 26-s + 29-s + 34-s + 37-s + 2·40-s − 41-s − 2·45-s − 50-s − 2·53-s − 58-s − 61-s + 64-s + 2·65-s − 72-s + 2·73-s − 74-s − 2·80-s + ⋯ |
| L(s) = 1 | − 2-s + 2·5-s + 8-s − 9-s − 2·10-s + 13-s − 16-s − 17-s + 18-s + 25-s − 26-s + 29-s + 34-s + 37-s + 2·40-s − 41-s − 2·45-s − 50-s − 2·53-s − 58-s − 61-s + 64-s + 2·65-s − 72-s + 2·73-s − 74-s − 2·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6492304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6492304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9483691662\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9483691662\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
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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
−9.134265686445477997466758034518, −9.109312922090248561125235224728, −8.602880915421017714090601559586, −8.293430408380406659709396652305, −7.88603282786073451744839800341, −7.59555450829677879950861721934, −6.84516238872075444258641221146, −6.52193107307509999959785619766, −6.14839445268312170742590196049, −6.03528529196262288817207027704, −5.43923495156482472704035592094, −5.07378625171470167812575069536, −4.58117094386622105048296908422, −4.25743084762675764148233833270, −3.33084443091194683963830048843, −3.17900611390416159734678392462, −2.16843254727874637038499768582, −2.15064108310085343127380531621, −1.55916518585881704397207929260, −0.78377813071204582965551940846,
0.78377813071204582965551940846, 1.55916518585881704397207929260, 2.15064108310085343127380531621, 2.16843254727874637038499768582, 3.17900611390416159734678392462, 3.33084443091194683963830048843, 4.25743084762675764148233833270, 4.58117094386622105048296908422, 5.07378625171470167812575069536, 5.43923495156482472704035592094, 6.03528529196262288817207027704, 6.14839445268312170742590196049, 6.52193107307509999959785619766, 6.84516238872075444258641221146, 7.59555450829677879950861721934, 7.88603282786073451744839800341, 8.293430408380406659709396652305, 8.602880915421017714090601559586, 9.109312922090248561125235224728, 9.134265686445477997466758034518
