| L(s) = 1 | − 2·2-s + 2·4-s − 5·5-s − 4·9-s + 10·10-s − 4·16-s − 7·17-s + 8·18-s − 10·20-s + 9·25-s − 29-s + 8·32-s + 14·34-s − 8·36-s + 8·37-s + 7·41-s + 20·45-s + 49-s − 18·50-s − 5·53-s + 2·58-s − 8·64-s − 14·68-s − 16·73-s − 16·74-s + 20·80-s + 7·81-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 2.23·5-s − 4/3·9-s + 3.16·10-s − 16-s − 1.69·17-s + 1.88·18-s − 2.23·20-s + 9/5·25-s − 0.185·29-s + 1.41·32-s + 2.40·34-s − 4/3·36-s + 1.31·37-s + 1.09·41-s + 2.98·45-s + 1/7·49-s − 2.54·50-s − 0.686·53-s + 0.262·58-s − 64-s − 1.69·68-s − 1.87·73-s − 1.85·74-s + 2.23·80-s + 7/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3728 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3728 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 233 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 11 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 156 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
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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
−11.83837427862951481200224178812, −11.47602925131215012250786876412, −11.13902532918224021550658977463, −10.72901216013503134990546436020, −9.699909088328357412780525051317, −8.992394262821415711487234424598, −8.561549714923159334951800275866, −8.017150458790311441876451292378, −7.58814185338759693012895415611, −6.90535193519782986000335142622, −5.97658993134620300610463396041, −4.59585277278112872320985025558, −3.98950056937746953878112991726, −2.69703804433430864487863243499, 0,
2.69703804433430864487863243499, 3.98950056937746953878112991726, 4.59585277278112872320985025558, 5.97658993134620300610463396041, 6.90535193519782986000335142622, 7.58814185338759693012895415611, 8.017150458790311441876451292378, 8.561549714923159334951800275866, 8.992394262821415711487234424598, 9.699909088328357412780525051317, 10.72901216013503134990546436020, 11.13902532918224021550658977463, 11.47602925131215012250786876412, 11.83837427862951481200224178812
