| L(s) = 1 | + 2·3-s − 2·5-s − 7-s + 3·9-s + 11-s − 7·13-s − 4·15-s + 3·17-s + 5·19-s − 2·21-s + 3·23-s + 3·25-s + 4·27-s − 29-s − 31-s + 2·33-s + 2·35-s + 37-s − 14·39-s + 7·41-s + 16·43-s − 6·45-s − 3·47-s + 7·49-s + 6·51-s − 4·53-s − 2·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s − 0.377·7-s + 9-s + 0.301·11-s − 1.94·13-s − 1.03·15-s + 0.727·17-s + 1.14·19-s − 0.436·21-s + 0.625·23-s + 3/5·25-s + 0.769·27-s − 0.185·29-s − 0.179·31-s + 0.348·33-s + 0.338·35-s + 0.164·37-s − 2.24·39-s + 1.09·41-s + 2.43·43-s − 0.894·45-s − 0.437·47-s + 49-s + 0.840·51-s − 0.549·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16160400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16160400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.093665842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.093665842\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 67 | $C_2$ | \( 1 - 16 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 7 T + 8 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 13 T + 98 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 11 T + 38 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 7 T - 48 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
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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
−8.523121869379670869261598197449, −8.226092511783942496929075882156, −7.77388080323204718278084139125, −7.57926087729692550025178697787, −7.27369704541936828360470978865, −7.09321140788080575918386489883, −6.61336029544474662903924863331, −6.07080960604687865365446256858, −5.51298566459986640142029710113, −5.31736138417466182660038346506, −4.63128744477594458141505365172, −4.49590997965160349904598855369, −4.00552501085272860820747205290, −3.48608425531599299679690638388, −3.16307681379710851525817851492, −2.88527970145881767384816990499, −2.19695657364563690658098170859, −2.05005755495356891937845234828, −0.841703966956516799376904572713, −0.74278018820985881227813629710,
0.74278018820985881227813629710, 0.841703966956516799376904572713, 2.05005755495356891937845234828, 2.19695657364563690658098170859, 2.88527970145881767384816990499, 3.16307681379710851525817851492, 3.48608425531599299679690638388, 4.00552501085272860820747205290, 4.49590997965160349904598855369, 4.63128744477594458141505365172, 5.31736138417466182660038346506, 5.51298566459986640142029710113, 6.07080960604687865365446256858, 6.61336029544474662903924863331, 7.09321140788080575918386489883, 7.27369704541936828360470978865, 7.57926087729692550025178697787, 7.77388080323204718278084139125, 8.226092511783942496929075882156, 8.523121869379670869261598197449
