| L(s) = 1 | − 2-s − 4-s + 3·8-s − 3·9-s − 11-s − 16-s − 14·17-s + 3·18-s + 22-s − 7·25-s − 5·32-s + 14·34-s + 3·36-s + 9·41-s + 43-s + 44-s + 8·49-s + 7·50-s + 2·59-s + 7·64-s − 18·67-s + 14·68-s − 9·72-s − 12·73-s + 9·81-s − 9·82-s − 11·83-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s − 9-s − 0.301·11-s − 1/4·16-s − 3.39·17-s + 0.707·18-s + 0.213·22-s − 7/5·25-s − 0.883·32-s + 2.40·34-s + 1/2·36-s + 1.40·41-s + 0.152·43-s + 0.150·44-s + 8/7·49-s + 0.989·50-s + 0.260·59-s + 7/8·64-s − 2.19·67-s + 1.69·68-s − 1.06·72-s − 1.40·73-s + 81-s − 0.993·82-s − 1.20·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 51 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 47 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−7.37926903233106281578644015169, −7.02700641393289474313744464974, −6.65300992921871390495988325159, −6.00370812254319962891624050482, −5.69672026012443918984842887018, −5.28882623615192997098727442487, −4.48286085487211551669146086494, −4.21882144145436540752214716583, −4.11787306297578325903004584135, −3.06324532395669765005725822487, −2.51532614195444001695233408119, −2.13422303356819755322315936030, −1.34211207438343845841711822104, 0, 0,
1.34211207438343845841711822104, 2.13422303356819755322315936030, 2.51532614195444001695233408119, 3.06324532395669765005725822487, 4.11787306297578325903004584135, 4.21882144145436540752214716583, 4.48286085487211551669146086494, 5.28882623615192997098727442487, 5.69672026012443918984842887018, 6.00370812254319962891624050482, 6.65300992921871390495988325159, 7.02700641393289474313744464974, 7.37926903233106281578644015169
