| L(s) = 1 | − 3·2-s − 2·3-s + 4·4-s + 6·6-s − 3·8-s + 3·9-s − 2·11-s − 8·12-s − 2·13-s + 3·16-s + 12·17-s − 9·18-s − 4·19-s + 6·22-s + 4·23-s + 6·24-s − 5·25-s + 6·26-s − 4·27-s − 16·29-s + 8·31-s − 6·32-s + 4·33-s − 36·34-s + 12·36-s − 10·37-s + 12·38-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.15·3-s + 2·4-s + 2.44·6-s − 1.06·8-s + 9-s − 0.603·11-s − 2.30·12-s − 0.554·13-s + 3/4·16-s + 2.91·17-s − 2.12·18-s − 0.917·19-s + 1.27·22-s + 0.834·23-s + 1.22·24-s − 25-s + 1.17·26-s − 0.769·27-s − 2.97·29-s + 1.43·31-s − 1.06·32-s + 0.696·33-s − 6.17·34-s + 2·36-s − 1.64·37-s + 1.94·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2614689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2614689 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3886243298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3886243298\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 16 T + 117 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 10 T + 79 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 82 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 125 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 142 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 16 T + 210 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−9.443169202973369641617311594021, −9.400317977913097710570593126020, −8.903886452063052013559117840168, −8.524649432781679522984719247707, −7.79411292934807206931554205359, −7.62819729944523810633617193586, −7.50606998763773376492684119385, −7.22462958634975854523566779290, −6.34905405829755616043889290647, −5.81786533971778180846308007372, −5.77257477189267214362641864231, −5.36621248845182162621140308491, −4.68621960502611556592229805233, −4.22868064781601819288147691951, −3.43532907550187378214860793328, −3.16495715504391030713362267660, −2.11404604697061054501098014160, −1.74450108306809813732576957408, −0.78149473645515618841021118899, −0.58023986852057187785248996683,
0.58023986852057187785248996683, 0.78149473645515618841021118899, 1.74450108306809813732576957408, 2.11404604697061054501098014160, 3.16495715504391030713362267660, 3.43532907550187378214860793328, 4.22868064781601819288147691951, 4.68621960502611556592229805233, 5.36621248845182162621140308491, 5.77257477189267214362641864231, 5.81786533971778180846308007372, 6.34905405829755616043889290647, 7.22462958634975854523566779290, 7.50606998763773376492684119385, 7.62819729944523810633617193586, 7.79411292934807206931554205359, 8.524649432781679522984719247707, 8.903886452063052013559117840168, 9.400317977913097710570593126020, 9.443169202973369641617311594021
