| L(s) = 1 | − 2·2-s + 2·4-s − 4·7-s + 5·9-s + 8·14-s − 4·16-s + 6·17-s − 10·18-s + 8·23-s − 8·28-s − 16·31-s + 8·32-s − 12·34-s + 10·36-s + 14·41-s − 16·46-s − 4·47-s − 2·49-s + 32·62-s − 20·63-s − 8·64-s + 12·68-s + 4·71-s − 2·73-s + 20·79-s + 16·81-s − 28·82-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 1.51·7-s + 5/3·9-s + 2.13·14-s − 16-s + 1.45·17-s − 2.35·18-s + 1.66·23-s − 1.51·28-s − 2.87·31-s + 1.41·32-s − 2.05·34-s + 5/3·36-s + 2.18·41-s − 2.35·46-s − 0.583·47-s − 2/7·49-s + 4.06·62-s − 2.51·63-s − 64-s + 1.45·68-s + 0.474·71-s − 0.234·73-s + 2.25·79-s + 16/9·81-s − 3.09·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6187037697\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6187037697\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 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 | $C_2^2$ | \( 1 - 125 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−12.63361182118974857993864039366, −12.47836247763051375924052962880, −11.50633075871204520736568592307, −11.04626555256960862482581360953, −10.41494099545228516649232105319, −10.29235140885648310606680400665, −9.510645286897577543213549760291, −9.255478317943019969668684261273, −9.231744007273789179764854062034, −8.145155672510948515172063660705, −7.50478665036111130350966263658, −7.35139083793185245367223042612, −6.77350446763589549996568158008, −6.23154921092807264196758968729, −5.38531404035271914030739881437, −4.65718064282621281230457445853, −3.72207287361453798376546723385, −3.22727289444975976883275377036, −1.97544887826466942453083827343, −0.934394675937933553012814828373,
0.934394675937933553012814828373, 1.97544887826466942453083827343, 3.22727289444975976883275377036, 3.72207287361453798376546723385, 4.65718064282621281230457445853, 5.38531404035271914030739881437, 6.23154921092807264196758968729, 6.77350446763589549996568158008, 7.35139083793185245367223042612, 7.50478665036111130350966263658, 8.145155672510948515172063660705, 9.231744007273789179764854062034, 9.255478317943019969668684261273, 9.510645286897577543213549760291, 10.29235140885648310606680400665, 10.41494099545228516649232105319, 11.04626555256960862482581360953, 11.50633075871204520736568592307, 12.47836247763051375924052962880, 12.63361182118974857993864039366
