| L(s) = 1 | − 3-s − 4·5-s − 2·9-s + 4·11-s + 4·15-s + 8·23-s + 3·25-s + 5·27-s − 6·31-s − 4·33-s − 6·37-s + 8·45-s − 6·49-s + 16·53-s − 16·55-s + 12·59-s + 14·67-s − 8·69-s − 24·71-s − 3·75-s + 81-s + 8·89-s + 6·93-s − 6·97-s − 8·99-s − 8·103-s + 6·111-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78·5-s − 2/3·9-s + 1.20·11-s + 1.03·15-s + 1.66·23-s + 3/5·25-s + 0.962·27-s − 1.07·31-s − 0.696·33-s − 0.986·37-s + 1.19·45-s − 6/7·49-s + 2.19·53-s − 2.15·55-s + 1.56·59-s + 1.71·67-s − 0.963·69-s − 2.84·71-s − 0.346·75-s + 1/9·81-s + 0.847·89-s + 0.622·93-s − 0.609·97-s − 0.804·99-s − 0.788·103-s + 0.569·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4460544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4460544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 11 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 9 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
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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
−7.12296835689531828419172314023, −6.96242379547504162594632274208, −6.43426161112425427802355654585, −5.83390414876252617735354391325, −5.60651088142050010085064931397, −4.96957496202027952333388779709, −4.74054722527680440231335392003, −4.07570044462901154585855245816, −3.69884500518693020828640564269, −3.53729039829921536298000164003, −2.89016245040251153046287074325, −2.25514668711721411393870235934, −1.39585341738655484686323839368, −0.73556239424125021006876266680, 0,
0.73556239424125021006876266680, 1.39585341738655484686323839368, 2.25514668711721411393870235934, 2.89016245040251153046287074325, 3.53729039829921536298000164003, 3.69884500518693020828640564269, 4.07570044462901154585855245816, 4.74054722527680440231335392003, 4.96957496202027952333388779709, 5.60651088142050010085064931397, 5.83390414876252617735354391325, 6.43426161112425427802355654585, 6.96242379547504162594632274208, 7.12296835689531828419172314023
