| L(s) = 1 | − 2·3-s + 4·5-s + 2·7-s + 2·9-s + 12·11-s + 2·13-s − 8·15-s + 2·17-s − 4·21-s − 10·23-s + 11·25-s − 6·27-s + 16·29-s − 24·33-s + 8·35-s + 10·37-s − 4·39-s − 12·41-s − 6·43-s + 8·45-s + 14·47-s + 2·49-s − 4·51-s − 2·53-s + 48·55-s + 4·63-s + 8·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.78·5-s + 0.755·7-s + 2/3·9-s + 3.61·11-s + 0.554·13-s − 2.06·15-s + 0.485·17-s − 0.872·21-s − 2.08·23-s + 11/5·25-s − 1.15·27-s + 2.97·29-s − 4.17·33-s + 1.35·35-s + 1.64·37-s − 0.640·39-s − 1.87·41-s − 0.914·43-s + 1.19·45-s + 2.04·47-s + 2/7·49-s − 0.560·51-s − 0.274·53-s + 6.47·55-s + 0.503·63-s + 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.612513031\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.612513031\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−10.00603090983227803823732108621, −9.553137061725573685451791805810, −9.087780250667918923722145217317, −8.820085653683374261958793737505, −8.339847627766752971299698340746, −7.991891818422554263370236683752, −7.02238575316111826498368085189, −6.81346533688698418485773298964, −6.44304473232919350683595808244, −6.13907769185402482137622622537, −5.68263435730118088526060662918, −5.65754511724089101590340936170, −4.59322451195134191619613165001, −4.51241501316971587164244080249, −3.96449958427301753893699282132, −3.41873533998996740940980024019, −2.49691984486803498770277463051, −1.79359317368167541320663463929, −1.27456796170607859190971806993, −1.10196067203519711290669230965,
1.10196067203519711290669230965, 1.27456796170607859190971806993, 1.79359317368167541320663463929, 2.49691984486803498770277463051, 3.41873533998996740940980024019, 3.96449958427301753893699282132, 4.51241501316971587164244080249, 4.59322451195134191619613165001, 5.65754511724089101590340936170, 5.68263435730118088526060662918, 6.13907769185402482137622622537, 6.44304473232919350683595808244, 6.81346533688698418485773298964, 7.02238575316111826498368085189, 7.991891818422554263370236683752, 8.339847627766752971299698340746, 8.820085653683374261958793737505, 9.087780250667918923722145217317, 9.553137061725573685451791805810, 10.00603090983227803823732108621
