| L(s) = 1 | + 2·5-s − 4·7-s + 2·9-s − 8·19-s + 3·25-s − 8·35-s − 4·37-s − 4·43-s + 4·45-s + 2·49-s + 12·53-s − 8·63-s − 5·81-s + 12·83-s + 4·89-s − 16·95-s − 12·97-s + 4·107-s − 28·113-s − 11·121-s + 4·125-s + 127-s + 131-s + 32·133-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.51·7-s + 2/3·9-s − 1.83·19-s + 3/5·25-s − 1.35·35-s − 0.657·37-s − 0.609·43-s + 0.596·45-s + 2/7·49-s + 1.64·53-s − 1.00·63-s − 5/9·81-s + 1.31·83-s + 0.423·89-s − 1.64·95-s − 1.21·97-s + 0.386·107-s − 2.63·113-s − 121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 2.77·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.614877366899767727212383683117, −8.020415365534980254960565516850, −7.42693142251341660894704744947, −6.73881553733037199099744681910, −6.65404010239088519836552140832, −6.26115860963273546153702682988, −5.67013069192851162699685276621, −5.17812631100724129492622947089, −4.52300468555237540728845461628, −3.92342525820680135983012290432, −3.48792670261330650874399245093, −2.66976939063195263823717000562, −2.21421865837896487879653154980, −1.35904199802600644391707963791, 0,
1.35904199802600644391707963791, 2.21421865837896487879653154980, 2.66976939063195263823717000562, 3.48792670261330650874399245093, 3.92342525820680135983012290432, 4.52300468555237540728845461628, 5.17812631100724129492622947089, 5.67013069192851162699685276621, 6.26115860963273546153702682988, 6.65404010239088519836552140832, 6.73881553733037199099744681910, 7.42693142251341660894704744947, 8.020415365534980254960565516850, 8.614877366899767727212383683117
