| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s − 2·5-s − 4·6-s − 4·8-s + 3·9-s + 4·10-s − 4·11-s + 6·12-s + 2·13-s − 4·15-s + 5·16-s − 8·17-s − 6·18-s + 10·19-s − 6·20-s + 8·22-s − 10·23-s − 8·24-s − 4·26-s + 4·27-s − 2·29-s + 8·30-s − 4·31-s − 6·32-s − 8·33-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.894·5-s − 1.63·6-s − 1.41·8-s + 9-s + 1.26·10-s − 1.20·11-s + 1.73·12-s + 0.554·13-s − 1.03·15-s + 5/4·16-s − 1.94·17-s − 1.41·18-s + 2.29·19-s − 1.34·20-s + 1.70·22-s − 2.08·23-s − 1.63·24-s − 0.784·26-s + 0.769·27-s − 0.371·29-s + 1.46·30-s − 0.718·31-s − 1.06·32-s − 1.39·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 19 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 43 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 10 T + 64 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 31 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 76 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 10 T + 100 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 75 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 31 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 22 T + 235 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 22 T + 260 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 16 T + 202 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 18 T + 196 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 180 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−8.137268305236160172415194446149, −8.075866292435197277483842580689, −7.62761906499602603133425477889, −7.61723073235035246849060056607, −6.91843924780574986112825004724, −6.89985659807353539749792378616, −6.24092644997960384654825484801, −5.70378190625531471709317989893, −5.51237617004546589237254956116, −4.83677546507183315512320937418, −4.17317481185997228729501704676, −4.14316911150518593870444560600, −3.36099736752790372719097445675, −3.18906377798137353750742715907, −2.56525921700966039264616764806, −2.28330578840492803100587690073, −1.63302423646835084744839110438, −1.24172395683391563791530561812, 0, 0,
1.24172395683391563791530561812, 1.63302423646835084744839110438, 2.28330578840492803100587690073, 2.56525921700966039264616764806, 3.18906377798137353750742715907, 3.36099736752790372719097445675, 4.14316911150518593870444560600, 4.17317481185997228729501704676, 4.83677546507183315512320937418, 5.51237617004546589237254956116, 5.70378190625531471709317989893, 6.24092644997960384654825484801, 6.89985659807353539749792378616, 6.91843924780574986112825004724, 7.61723073235035246849060056607, 7.62761906499602603133425477889, 8.075866292435197277483842580689, 8.137268305236160172415194446149
