| L(s) = 1 | − 4·7-s − 6·9-s + 8·19-s + 25-s − 4·29-s − 16·31-s + 12·37-s + 8·47-s + 9·49-s + 12·53-s − 8·59-s + 24·63-s + 27·81-s − 32·83-s + 8·103-s + 28·109-s + 36·113-s − 6·121-s + 127-s + 131-s − 32·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 2·9-s + 1.83·19-s + 1/5·25-s − 0.742·29-s − 2.87·31-s + 1.97·37-s + 1.16·47-s + 9/7·49-s + 1.64·53-s − 1.04·59-s + 3.02·63-s + 3·81-s − 3.51·83-s + 0.788·103-s + 2.68·109-s + 3.38·113-s − 0.545·121-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 + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8328373048\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8328373048\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−9.060844997255098876416985881509, −9.022035638415746178025932660186, −8.454267864352545795806495542150, −7.67375816764907867780022374833, −7.28236915503800031844503481655, −6.97238544391138654785050907556, −5.93296579879359794523720946560, −5.71297380596599560620236986392, −5.70093888866808737514223874840, −4.69948739897509711013983659515, −3.79782147759071587315933837414, −3.26180587408034592610487247010, −2.95971176692665828505622566750, −2.14273353770490399369013857078, −0.58317500635071871496421307783,
0.58317500635071871496421307783, 2.14273353770490399369013857078, 2.95971176692665828505622566750, 3.26180587408034592610487247010, 3.79782147759071587315933837414, 4.69948739897509711013983659515, 5.70093888866808737514223874840, 5.71297380596599560620236986392, 5.93296579879359794523720946560, 6.97238544391138654785050907556, 7.28236915503800031844503481655, 7.67375816764907867780022374833, 8.454267864352545795806495542150, 9.022035638415746178025932660186, 9.060844997255098876416985881509
