| L(s) = 1 | − 4-s + 2·7-s − 2·9-s + 6·11-s + 4·13-s − 3·16-s − 4·25-s − 2·28-s + 2·36-s + 4·37-s − 2·43-s − 6·44-s + 18·47-s + 3·49-s − 4·52-s − 6·53-s + 18·59-s − 4·63-s + 7·64-s + 12·77-s − 5·81-s − 18·89-s + 8·91-s − 2·97-s − 12·99-s + 4·100-s − 18·107-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.755·7-s − 2/3·9-s + 1.80·11-s + 1.10·13-s − 3/4·16-s − 4/5·25-s − 0.377·28-s + 1/3·36-s + 0.657·37-s − 0.304·43-s − 0.904·44-s + 2.62·47-s + 3/7·49-s − 0.554·52-s − 0.824·53-s + 2.34·59-s − 0.503·63-s + 7/8·64-s + 1.36·77-s − 5/9·81-s − 1.90·89-s + 0.838·91-s − 0.203·97-s − 1.20·99-s + 2/5·100-s − 1.74·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 137641 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137641 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.669195602\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.669195602\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 53 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 88 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−9.219937111405886687938224272873, −8.875997361435725407718270053947, −8.413683038907556054252999052743, −8.169469807034162174579554322705, −7.30704178772340917409702952693, −6.91127889968299916133971849368, −6.30904577818241369847981586875, −5.80096868393801734660086363232, −5.39559530103606270776732284952, −4.52963928910530351398167601494, −4.02881561405709240393387605390, −3.79160819475929835705265270358, −2.74539152112881664366048989592, −1.86379800540961638344508185450, −0.971514613725023959310157844924,
0.971514613725023959310157844924, 1.86379800540961638344508185450, 2.74539152112881664366048989592, 3.79160819475929835705265270358, 4.02881561405709240393387605390, 4.52963928910530351398167601494, 5.39559530103606270776732284952, 5.80096868393801734660086363232, 6.30904577818241369847981586875, 6.91127889968299916133971849368, 7.30704178772340917409702952693, 8.169469807034162174579554322705, 8.413683038907556054252999052743, 8.875997361435725407718270053947, 9.219937111405886687938224272873
