| L(s) = 1 | − 2·3-s + 8·7-s + 9-s − 2·19-s − 16·21-s + 8·25-s + 4·27-s − 12·29-s − 4·43-s + 34·49-s + 12·53-s + 4·57-s − 24·59-s + 4·61-s + 8·63-s + 24·71-s − 8·73-s − 16·75-s − 11·81-s + 24·87-s + 12·89-s + 24·107-s − 24·113-s − 10·121-s + 127-s + 8·129-s + 131-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 3.02·7-s + 1/3·9-s − 0.458·19-s − 3.49·21-s + 8/5·25-s + 0.769·27-s − 2.22·29-s − 0.609·43-s + 34/7·49-s + 1.64·53-s + 0.529·57-s − 3.12·59-s + 0.512·61-s + 1.00·63-s + 2.84·71-s − 0.936·73-s − 1.84·75-s − 1.22·81-s + 2.57·87-s + 1.27·89-s + 2.32·107-s − 2.25·113-s − 0.909·121-s + 0.0887·127-s + 0.704·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.881532613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.881532613\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 19 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 92 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 140 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
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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
−10.40998564862474310029535041263, −10.25431279141766577006478170612, −9.216413834298130363797134814701, −9.123330094805560861564768149051, −8.491324329595568610739341171650, −8.250080617653198518090564252988, −7.62979132577271948193597461807, −7.56829973829297645644234149581, −6.84969212935742716411216173767, −6.45485779046863117870350817663, −5.74309766874479193366556730008, −5.44278028212658796088305954183, −4.96714253510858888981162347425, −4.80817025560528088191897952505, −4.26975062307613457678442206287, −3.68131958617409451221659791772, −2.75006248232345300893983611783, −1.92410037831833731011214188563, −1.59728233417927501536077642749, −0.74710698487958906672737704031,
0.74710698487958906672737704031, 1.59728233417927501536077642749, 1.92410037831833731011214188563, 2.75006248232345300893983611783, 3.68131958617409451221659791772, 4.26975062307613457678442206287, 4.80817025560528088191897952505, 4.96714253510858888981162347425, 5.44278028212658796088305954183, 5.74309766874479193366556730008, 6.45485779046863117870350817663, 6.84969212935742716411216173767, 7.56829973829297645644234149581, 7.62979132577271948193597461807, 8.250080617653198518090564252988, 8.491324329595568610739341171650, 9.123330094805560861564768149051, 9.216413834298130363797134814701, 10.25431279141766577006478170612, 10.40998564862474310029535041263
