| L(s) = 1 | − 2·2-s + 2·4-s + 12·13-s − 4·16-s − 24·26-s − 16·31-s + 8·32-s + 4·37-s − 14·41-s − 8·43-s + 10·49-s + 24·52-s + 8·53-s + 32·62-s − 8·64-s − 6·67-s − 4·71-s − 8·74-s − 20·79-s + 28·82-s + 18·83-s + 16·86-s + 10·89-s − 20·98-s − 16·106-s − 34·107-s − 3·121-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 3.32·13-s − 16-s − 4.70·26-s − 2.87·31-s + 1.41·32-s + 0.657·37-s − 2.18·41-s − 1.21·43-s + 10/7·49-s + 3.32·52-s + 1.09·53-s + 4.06·62-s − 64-s − 0.733·67-s − 0.474·71-s − 0.929·74-s − 2.25·79-s + 3.09·82-s + 1.97·83-s + 1.72·86-s + 1.05·89-s − 2.02·98-s − 1.55·106-s − 3.28·107-s − 0.272·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.031172949\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.031172949\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−9.272346907461526443367500875955, −8.940168064701903361975486213626, −8.819557507877414929633644689804, −8.344399699853865830503421916963, −8.147281054666318076395470561938, −7.64048312778420507447075199579, −7.15802248013953371174448335116, −6.78723493375812161301749780699, −6.45028731803533446933874218147, −5.97465032313762519018002891526, −5.43701622705332516190092119423, −5.30943316171133148818326756496, −4.22944591968585593045598515101, −4.11544894245825634381765244018, −3.41769203788861015066601730553, −3.25995214970221879041770513945, −2.23215586395140745829944404859, −1.55532738366637972648766353571, −1.43076874910422257869030661404, −0.51594823997442539477854728728,
0.51594823997442539477854728728, 1.43076874910422257869030661404, 1.55532738366637972648766353571, 2.23215586395140745829944404859, 3.25995214970221879041770513945, 3.41769203788861015066601730553, 4.11544894245825634381765244018, 4.22944591968585593045598515101, 5.30943316171133148818326756496, 5.43701622705332516190092119423, 5.97465032313762519018002891526, 6.45028731803533446933874218147, 6.78723493375812161301749780699, 7.15802248013953371174448335116, 7.64048312778420507447075199579, 8.147281054666318076395470561938, 8.344399699853865830503421916963, 8.819557507877414929633644689804, 8.940168064701903361975486213626, 9.272346907461526443367500875955
