| L(s) = 1 | + 2·5-s + 4·7-s − 4·11-s − 2·13-s + 4·17-s − 4·19-s + 8·23-s + 3·25-s + 8·29-s − 4·31-s + 8·35-s + 12·41-s + 8·43-s − 4·47-s − 2·49-s + 12·53-s − 8·55-s + 4·59-s + 8·61-s − 4·65-s − 20·67-s − 12·71-s − 16·77-s − 9·81-s + 4·83-s + 8·85-s + 12·89-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.51·7-s − 1.20·11-s − 0.554·13-s + 0.970·17-s − 0.917·19-s + 1.66·23-s + 3/5·25-s + 1.48·29-s − 0.718·31-s + 1.35·35-s + 1.87·41-s + 1.21·43-s − 0.583·47-s − 2/7·49-s + 1.64·53-s − 1.07·55-s + 0.520·59-s + 1.02·61-s − 0.496·65-s − 2.44·67-s − 1.42·71-s − 1.82·77-s − 81-s + 0.439·83-s + 0.867·85-s + 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.480008842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.480008842\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 36 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 56 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 48 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 68 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 20 T + 210 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 172 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 134 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−10.81735829551496376384178067299, −10.70525758672027089757299013829, −10.35473728271326370463691662290, −9.855490631365725808039136213883, −9.297786020336508059903442128787, −8.828004100374299851941654892946, −8.468772761594242766495781029739, −7.949433548834261159646894082631, −7.45021289498554912174980419510, −7.24832421898336060270397296698, −6.45042225654760342330317057616, −5.90694615766181500994260288707, −5.38833856141713142480185624416, −5.04198320919738406846616321643, −4.62367180616311040522304514495, −4.01831226697534092421927954877, −2.77598347198424651725791736953, −2.72510253910502481469015184697, −1.80293423917785422604569313169, −1.01665628423807691467792718144,
1.01665628423807691467792718144, 1.80293423917785422604569313169, 2.72510253910502481469015184697, 2.77598347198424651725791736953, 4.01831226697534092421927954877, 4.62367180616311040522304514495, 5.04198320919738406846616321643, 5.38833856141713142480185624416, 5.90694615766181500994260288707, 6.45042225654760342330317057616, 7.24832421898336060270397296698, 7.45021289498554912174980419510, 7.949433548834261159646894082631, 8.468772761594242766495781029739, 8.828004100374299851941654892946, 9.297786020336508059903442128787, 9.855490631365725808039136213883, 10.35473728271326370463691662290, 10.70525758672027089757299013829, 10.81735829551496376384178067299
