| L(s) = 1 | + 2·2-s + 3·4-s − 4·7-s + 4·8-s − 2·9-s + 12·11-s − 8·14-s + 5·16-s − 4·18-s + 24·22-s − 10·25-s − 12·28-s + 6·32-s − 6·36-s − 8·37-s + 16·43-s + 36·44-s + 9·49-s − 20·50-s − 12·53-s − 16·56-s + 8·63-s + 7·64-s + 16·67-s − 8·72-s − 16·74-s − 48·77-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 1.51·7-s + 1.41·8-s − 2/3·9-s + 3.61·11-s − 2.13·14-s + 5/4·16-s − 0.942·18-s + 5.11·22-s − 2·25-s − 2.26·28-s + 1.06·32-s − 36-s − 1.31·37-s + 2.43·43-s + 5.42·44-s + 9/7·49-s − 2.82·50-s − 1.64·53-s − 2.13·56-s + 1.00·63-s + 7/8·64-s + 1.95·67-s − 0.942·72-s − 1.85·74-s − 5.47·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56644 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56644 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.167608159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.167608159\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−9.795849732674491714602252272054, −9.435449454754526833910951528272, −9.286071988264369821451329617999, −8.550482821819457584609282080376, −7.75961744967946030738916282437, −6.98368217520104403156128027372, −6.65330731261113455462227374582, −6.16055468076817207109148001002, −5.99911855171913780844071238816, −5.16488039277883802975012518677, −4.04100934533934329070442787144, −3.90229547122613769308947697962, −3.50747685387131529534531471772, −2.52652883309992049334898398402, −1.44781063014612234378860428910,
1.44781063014612234378860428910, 2.52652883309992049334898398402, 3.50747685387131529534531471772, 3.90229547122613769308947697962, 4.04100934533934329070442787144, 5.16488039277883802975012518677, 5.99911855171913780844071238816, 6.16055468076817207109148001002, 6.65330731261113455462227374582, 6.98368217520104403156128027372, 7.75961744967946030738916282437, 8.550482821819457584609282080376, 9.286071988264369821451329617999, 9.435449454754526833910951528272, 9.795849732674491714602252272054
