| L(s) = 1 | − 3·3-s − 4·7-s + 6·9-s + 6·17-s + 12·21-s − 9·27-s + 16·37-s − 12·41-s − 20·43-s + 6·47-s + 9·49-s − 18·51-s − 12·59-s − 24·63-s − 4·67-s − 26·79-s + 9·81-s − 24·83-s − 36·101-s − 22·109-s − 48·111-s − 24·119-s − 5·121-s + 36·123-s + 127-s + 60·129-s + 131-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.51·7-s + 2·9-s + 1.45·17-s + 2.61·21-s − 1.73·27-s + 2.63·37-s − 1.87·41-s − 3.04·43-s + 0.875·47-s + 9/7·49-s − 2.52·51-s − 1.56·59-s − 3.02·63-s − 0.488·67-s − 2.92·79-s + 81-s − 2.63·83-s − 3.58·101-s − 2.10·109-s − 4.55·111-s − 2.20·119-s − 0.454·121-s + 3.24·123-s + 0.0887·127-s + 5.28·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1249831536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1249831536\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 191 T^{2} + p^{2} T^{4} \) |
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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
−9.729184435971194510321989448901, −9.008262180957748152244164919088, −8.463213498431155976363988716114, −8.127657355138364789031369488175, −7.61912106993903455734081943332, −7.04672926578760164840412656180, −6.99258156101152089273126480666, −6.39270562968678152146015670687, −6.20581759718519006484921154544, −5.73756902911413393800885070685, −5.46660171347011354232674929558, −5.02935636987606817124427535089, −4.50769752810208865971432663105, −4.04536584362024925044513549539, −3.60783018052479318317420373352, −2.85584658164941349263425279384, −2.83211408681181533885687999866, −1.41982939658653832192944891929, −1.33488803561222162646106126109, −0.15278607508581560021782569393,
0.15278607508581560021782569393, 1.33488803561222162646106126109, 1.41982939658653832192944891929, 2.83211408681181533885687999866, 2.85584658164941349263425279384, 3.60783018052479318317420373352, 4.04536584362024925044513549539, 4.50769752810208865971432663105, 5.02935636987606817124427535089, 5.46660171347011354232674929558, 5.73756902911413393800885070685, 6.20581759718519006484921154544, 6.39270562968678152146015670687, 6.99258156101152089273126480666, 7.04672926578760164840412656180, 7.61912106993903455734081943332, 8.127657355138364789031369488175, 8.463213498431155976363988716114, 9.008262180957748152244164919088, 9.729184435971194510321989448901
