| L(s) = 1 | + 3-s + 9-s − 11-s − 8·17-s − 6·25-s + 27-s + 16·29-s − 16·31-s − 33-s + 20·37-s − 16·41-s − 10·49-s − 8·51-s + 24·67-s − 6·75-s + 81-s − 32·83-s + 16·87-s − 16·93-s − 4·97-s − 99-s + 32·101-s + 8·103-s + 20·111-s + 121-s − 16·123-s + 127-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.301·11-s − 1.94·17-s − 6/5·25-s + 0.192·27-s + 2.97·29-s − 2.87·31-s − 0.174·33-s + 3.28·37-s − 2.49·41-s − 1.42·49-s − 1.12·51-s + 2.93·67-s − 0.692·75-s + 1/9·81-s − 3.51·83-s + 1.71·87-s − 1.65·93-s − 0.406·97-s − 0.100·99-s + 3.18·101-s + 0.788·103-s + 1.89·111-s + 1/11·121-s − 1.44·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 574992 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 574992 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 - T \) |
| 11 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.071554028904627902844738284977, −8.069373725115527033692000658280, −7.13458477543704272109197550389, −7.05430755180880074285547369845, −6.25340607782487029859951366723, −6.20097915233518496322100006686, −5.33648935965715263647979723486, −4.75118710346056158176317605927, −4.50322513346415393072595559425, −3.79814948526845934439056144160, −3.35247680087092108642086649384, −2.46483616139600209242493840583, −2.26769467555400519962633969149, −1.33575734148260048331554449892, 0,
1.33575734148260048331554449892, 2.26769467555400519962633969149, 2.46483616139600209242493840583, 3.35247680087092108642086649384, 3.79814948526845934439056144160, 4.50322513346415393072595559425, 4.75118710346056158176317605927, 5.33648935965715263647979723486, 6.20097915233518496322100006686, 6.25340607782487029859951366723, 7.05430755180880074285547369845, 7.13458477543704272109197550389, 8.069373725115527033692000658280, 8.071554028904627902844738284977
