| L(s) = 1 | − 2·2-s + 2·3-s + 2·4-s − 4·5-s − 4·6-s + 3·9-s + 8·10-s + 4·12-s − 12·13-s − 8·15-s − 4·16-s − 6·18-s − 8·20-s + 11·25-s + 24·26-s + 4·27-s + 16·30-s − 16·31-s + 8·32-s + 6·36-s − 4·37-s − 24·39-s − 16·41-s − 12·43-s − 12·45-s − 8·48-s − 49-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 4-s − 1.78·5-s − 1.63·6-s + 9-s + 2.52·10-s + 1.15·12-s − 3.32·13-s − 2.06·15-s − 16-s − 1.41·18-s − 1.78·20-s + 11/5·25-s + 4.70·26-s + 0.769·27-s + 2.92·30-s − 2.87·31-s + 1.41·32-s + 36-s − 0.657·37-s − 3.84·39-s − 2.49·41-s − 1.82·43-s − 1.78·45-s − 1.15·48-s − 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−9.827587621863316540391035609981, −9.524576000525702766417180619032, −8.938844465520380847061105951189, −8.926696698067073916940521242340, −8.076355883728250884671638837610, −8.045482269095616553945119907987, −7.49802308981296106456263057498, −7.35913607684984123440333953142, −6.85490533945410981959665742012, −6.70366099803928149041269369318, −5.26565231921234808176732281286, −4.95540436795898345333706998985, −4.62861318454356888034716074040, −3.91565065162201978974311969057, −3.31940275563986170186367689426, −2.96577385877494264927894341267, −2.07327500427967950594078941496, −1.71665184902814543981883396650, 0, 0,
1.71665184902814543981883396650, 2.07327500427967950594078941496, 2.96577385877494264927894341267, 3.31940275563986170186367689426, 3.91565065162201978974311969057, 4.62861318454356888034716074040, 4.95540436795898345333706998985, 5.26565231921234808176732281286, 6.70366099803928149041269369318, 6.85490533945410981959665742012, 7.35913607684984123440333953142, 7.49802308981296106456263057498, 8.045482269095616553945119907987, 8.076355883728250884671638837610, 8.926696698067073916940521242340, 8.938844465520380847061105951189, 9.524576000525702766417180619032, 9.827587621863316540391035609981
