| L(s) = 1 | + 3-s + 4-s − 8·5-s + 9-s + 12-s − 8·15-s + 16-s − 8·20-s + 38·25-s + 27-s + 36-s − 12·37-s + 8·43-s − 8·45-s + 16·47-s + 48-s − 8·59-s − 8·60-s + 64-s + 8·67-s + 38·75-s − 16·79-s − 8·80-s + 81-s + 24·83-s − 16·89-s + 38·100-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s − 3.57·5-s + 1/3·9-s + 0.288·12-s − 2.06·15-s + 1/4·16-s − 1.78·20-s + 38/5·25-s + 0.192·27-s + 1/6·36-s − 1.97·37-s + 1.21·43-s − 1.19·45-s + 2.33·47-s + 0.144·48-s − 1.04·59-s − 1.03·60-s + 1/8·64-s + 0.977·67-s + 4.38·75-s − 1.80·79-s − 0.894·80-s + 1/9·81-s + 2.63·83-s − 1.69·89-s + 19/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259308 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259308 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 - T \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−8.367002712496184399269342005462, −8.236567804687403362610493554923, −7.69885140727101440087412366746, −7.45554887409285403711020085491, −6.94059953288016050503047892713, −6.77970713278050838983490323162, −5.74587009674352948543449406475, −5.02209723965846660973920127041, −4.45230932233748260088756468320, −3.87915627242789051687615484994, −3.77980943086478052006597909549, −3.09526891884004259957229060750, −2.52889971936759396667083662829, −1.10984321000633935751921154525, 0,
1.10984321000633935751921154525, 2.52889971936759396667083662829, 3.09526891884004259957229060750, 3.77980943086478052006597909549, 3.87915627242789051687615484994, 4.45230932233748260088756468320, 5.02209723965846660973920127041, 5.74587009674352948543449406475, 6.77970713278050838983490323162, 6.94059953288016050503047892713, 7.45554887409285403711020085491, 7.69885140727101440087412366746, 8.236567804687403362610493554923, 8.367002712496184399269342005462
