| L(s) = 1 | + 4-s + 4·5-s − 6·9-s − 11-s + 16-s + 4·20-s − 16·23-s + 2·25-s − 16·31-s − 6·36-s − 4·37-s − 44-s − 24·45-s + 16·47-s + 49-s + 12·53-s − 4·55-s + 64-s − 24·67-s + 32·71-s + 4·80-s + 27·81-s − 12·89-s − 16·92-s + 20·97-s + 6·99-s + 2·100-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.78·5-s − 2·9-s − 0.301·11-s + 1/4·16-s + 0.894·20-s − 3.33·23-s + 2/5·25-s − 2.87·31-s − 36-s − 0.657·37-s − 0.150·44-s − 3.57·45-s + 2.33·47-s + 1/7·49-s + 1.64·53-s − 0.539·55-s + 1/8·64-s − 2.93·67-s + 3.79·71-s + 0.447·80-s + 3·81-s − 1.27·89-s − 1.66·92-s + 2.03·97-s + 0.603·99-s + 1/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260876 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260876 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−8.919520401542258285479247886410, −8.298613740465759678521949084161, −7.58187479426412561326624344048, −7.50564216619595368078781408441, −6.52101115191644800162484652268, −6.02643591301064267650899026350, −5.78813651063738143488715061532, −5.60617960823028323278478834583, −5.08084944453219704790793287622, −3.78476385235783926950251020606, −3.72195851889949585255191314251, −2.45595284593410008763714345701, −2.30984363521644281586554494696, −1.78687930018691785193380503928, 0,
1.78687930018691785193380503928, 2.30984363521644281586554494696, 2.45595284593410008763714345701, 3.72195851889949585255191314251, 3.78476385235783926950251020606, 5.08084944453219704790793287622, 5.60617960823028323278478834583, 5.78813651063738143488715061532, 6.02643591301064267650899026350, 6.52101115191644800162484652268, 7.50564216619595368078781408441, 7.58187479426412561326624344048, 8.298613740465759678521949084161, 8.919520401542258285479247886410
