| L(s) = 1 | + 3-s − 3·4-s + 9-s − 3·12-s + 5·16-s − 4·17-s + 27-s − 3·36-s + 20·37-s + 20·41-s − 8·43-s − 16·47-s + 5·48-s − 7·49-s − 4·51-s − 8·59-s − 3·64-s − 24·67-s + 12·68-s + 81-s − 24·83-s − 12·89-s + 12·101-s − 3·108-s + 28·109-s + 20·111-s − 6·121-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 3/2·4-s + 1/3·9-s − 0.866·12-s + 5/4·16-s − 0.970·17-s + 0.192·27-s − 1/2·36-s + 3.28·37-s + 3.12·41-s − 1.21·43-s − 2.33·47-s + 0.721·48-s − 49-s − 0.560·51-s − 1.04·59-s − 3/8·64-s − 2.93·67-s + 1.45·68-s + 1/9·81-s − 2.63·83-s − 1.27·89-s + 1.19·101-s − 0.288·108-s + 2.68·109-s + 1.89·111-s − 0.545·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 826875 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 826875 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( 1 - T \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.023793216333534711017938369158, −7.68003844385765985083943135134, −7.39848909917701976603786239429, −6.57558779208505392879804927456, −6.12621854048128780982239967429, −5.86497653842965752491328535183, −5.05351399151288949834601255152, −4.52160444884874669934496061646, −4.40145701518461031613845247878, −3.98361923014175761679111177485, −3.01549784140686353642765162463, −2.88208540738665705106266854904, −1.89845619892689055490680198063, −1.06721576698113432176731915153, 0,
1.06721576698113432176731915153, 1.89845619892689055490680198063, 2.88208540738665705106266854904, 3.01549784140686353642765162463, 3.98361923014175761679111177485, 4.40145701518461031613845247878, 4.52160444884874669934496061646, 5.05351399151288949834601255152, 5.86497653842965752491328535183, 6.12621854048128780982239967429, 6.57558779208505392879804927456, 7.39848909917701976603786239429, 7.68003844385765985083943135134, 8.023793216333534711017938369158
