| L(s) = 1 | + 2·5-s − 6·9-s + 4·11-s + 8·23-s + 3·25-s − 16·31-s + 12·37-s − 12·45-s + 8·47-s + 2·49-s + 12·53-s + 8·55-s − 8·59-s + 16·67-s + 27·81-s − 12·89-s − 28·97-s − 24·99-s + 8·103-s + 36·113-s + 16·115-s + 5·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 2·9-s + 1.20·11-s + 1.66·23-s + 3/5·25-s − 2.87·31-s + 1.97·37-s − 1.78·45-s + 1.16·47-s + 2/7·49-s + 1.64·53-s + 1.07·55-s − 1.04·59-s + 1.95·67-s + 3·81-s − 1.27·89-s − 2.84·97-s − 2.41·99-s + 0.788·103-s + 3.38·113-s + 1.49·115-s + 5/11·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.807192052\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.807192052\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−9.022035638415746178025932660186, −8.903529359998922904579331677853, −8.319898091581096916744910484661, −7.70303958223137845782045497044, −6.97238544391138654785050907556, −6.81644823665973489085396145114, −5.94028444997569064994570219738, −5.70093888866808737514223874840, −5.41810682445711458041939493554, −4.62487051901560546933133606596, −3.87138095586879394292498913230, −3.26180587408034592610487247010, −2.64264527642459335277526208075, −1.99873426678938511502281122159, −0.880315059374556524645724595724,
0.880315059374556524645724595724, 1.99873426678938511502281122159, 2.64264527642459335277526208075, 3.26180587408034592610487247010, 3.87138095586879394292498913230, 4.62487051901560546933133606596, 5.41810682445711458041939493554, 5.70093888866808737514223874840, 5.94028444997569064994570219738, 6.81644823665973489085396145114, 6.97238544391138654785050907556, 7.70303958223137845782045497044, 8.319898091581096916744910484661, 8.903529359998922904579331677853, 9.022035638415746178025932660186
