| L(s) = 1 | − 2·3-s − 3·5-s + 2·7-s + 3·9-s + 2·11-s + 2·13-s + 6·15-s − 6·17-s − 19-s − 4·21-s − 23-s + 25-s − 4·27-s − 11·29-s + 7·31-s − 4·33-s − 6·35-s − 6·37-s − 4·39-s − 4·41-s − 43-s − 9·45-s + 5·47-s + 3·49-s + 12·51-s − 15·53-s − 6·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.34·5-s + 0.755·7-s + 9-s + 0.603·11-s + 0.554·13-s + 1.54·15-s − 1.45·17-s − 0.229·19-s − 0.872·21-s − 0.208·23-s + 1/5·25-s − 0.769·27-s − 2.04·29-s + 1.25·31-s − 0.696·33-s − 1.01·35-s − 0.986·37-s − 0.640·39-s − 0.624·41-s − 0.152·43-s − 1.34·45-s + 0.729·47-s + 3/7·49-s + 1.68·51-s − 2.06·53-s − 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19079424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19079424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + T + 34 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + T + 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 11 T + 84 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 7 T + 70 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + T + 82 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 5 T + 62 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 15 T + 124 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 10 T + 142 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 5 T + 148 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 11 T + 150 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 13 T + 170 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 9 T + 160 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 17 T + 228 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
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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
−8.081323220541033722847345244839, −7.87575271323689612241316906679, −7.44667140030201639559076286664, −7.02318286676874751154479085358, −6.68739668382502249755221281317, −6.50379174336822204097847003745, −5.96973163612328114103038614432, −5.62690127594722911619964936076, −5.03209765246110182136305336756, −4.98140491666561874410112530414, −4.23562831777664132389766483216, −4.20616277974553578246526970616, −3.63419138205100958635572851651, −3.56315743509505356394725820019, −2.54677968075550301463800410797, −2.13537448475490912460453031786, −1.50015831329005986987306890297, −1.08444517067964097977084642172, 0, 0,
1.08444517067964097977084642172, 1.50015831329005986987306890297, 2.13537448475490912460453031786, 2.54677968075550301463800410797, 3.56315743509505356394725820019, 3.63419138205100958635572851651, 4.20616277974553578246526970616, 4.23562831777664132389766483216, 4.98140491666561874410112530414, 5.03209765246110182136305336756, 5.62690127594722911619964936076, 5.96973163612328114103038614432, 6.50379174336822204097847003745, 6.68739668382502249755221281317, 7.02318286676874751154479085358, 7.44667140030201639559076286664, 7.87575271323689612241316906679, 8.081323220541033722847345244839
