| L(s) = 1 | + 3·4-s − 9-s + 8·11-s + 5·16-s + 20·29-s + 8·31-s − 3·36-s + 12·41-s + 24·44-s − 2·49-s − 24·59-s − 4·61-s + 3·64-s − 16·79-s + 81-s + 4·89-s − 8·99-s − 36·101-s + 4·109-s + 60·116-s + 26·121-s + 24·124-s + 127-s + 131-s + 137-s + 139-s − 5·144-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 1/3·9-s + 2.41·11-s + 5/4·16-s + 3.71·29-s + 1.43·31-s − 1/2·36-s + 1.87·41-s + 3.61·44-s − 2/7·49-s − 3.12·59-s − 0.512·61-s + 3/8·64-s − 1.80·79-s + 1/9·81-s + 0.423·89-s − 0.804·99-s − 3.58·101-s + 0.383·109-s + 5.57·116-s + 2.36·121-s + 2.15·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.416·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.182790063\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.182790063\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−10.25800440084270862727443334908, −9.843949731345488666895966813259, −9.330644841068273565374587501202, −9.074120607716495302519589436104, −8.394276042197933400959604611790, −8.259961352261985330843943341673, −7.62175974472338047337191065193, −7.16356864113273012867168597601, −6.60772784966033004703867759263, −6.44976994964412690127017763482, −6.21070364588504495769353006399, −5.76021861845165782968413195257, −4.82456405177212311162922531074, −4.40552883510858598627716815176, −4.08240470883098429283240900463, −3.05977221831161837048319441795, −2.97693954633555855295601697523, −2.33495709717199920067582876212, −1.24989093101712286932727143464, −1.23079341473238165615885235245,
1.23079341473238165615885235245, 1.24989093101712286932727143464, 2.33495709717199920067582876212, 2.97693954633555855295601697523, 3.05977221831161837048319441795, 4.08240470883098429283240900463, 4.40552883510858598627716815176, 4.82456405177212311162922531074, 5.76021861845165782968413195257, 6.21070364588504495769353006399, 6.44976994964412690127017763482, 6.60772784966033004703867759263, 7.16356864113273012867168597601, 7.62175974472338047337191065193, 8.259961352261985330843943341673, 8.394276042197933400959604611790, 9.074120607716495302519589436104, 9.330644841068273565374587501202, 9.843949731345488666895966813259, 10.25800440084270862727443334908
