| L(s) = 1 | − 2·5-s − 3·7-s + 3·11-s − 2·13-s − 5·17-s + 3·23-s + 3·25-s + 4·31-s + 6·35-s − 3·37-s − 5·41-s + 4·43-s + 20·47-s − 3·49-s + 11·53-s − 6·55-s + 14·59-s − 9·61-s + 4·65-s − 6·67-s − 9·71-s − 12·73-s − 9·77-s + 5·79-s − 6·83-s + 10·85-s + 5·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.13·7-s + 0.904·11-s − 0.554·13-s − 1.21·17-s + 0.625·23-s + 3/5·25-s + 0.718·31-s + 1.01·35-s − 0.493·37-s − 0.780·41-s + 0.609·43-s + 2.91·47-s − 3/7·49-s + 1.51·53-s − 0.809·55-s + 1.82·59-s − 1.15·61-s + 0.496·65-s − 0.733·67-s − 1.06·71-s − 1.40·73-s − 1.02·77-s + 0.562·79-s − 0.658·83-s + 1.08·85-s + 0.529·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 3 T + 12 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 3 T + 72 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 5 T + 84 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 11 T + 132 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 14 T + 150 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 9 T + 104 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 126 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 9 T + 56 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 5 T + 58 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 158 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 5 T - 24 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 9 T + 108 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
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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
−7.29576348138336129242005600587, −7.22428003356006791214523657559, −6.79376092553121232131899446021, −6.77605196387338580151876308122, −6.11454383503353829474283256633, −6.01365968308380022181433890173, −5.45039613103266583301269945839, −5.17621653249316015149339453638, −4.45234389988403752872465081444, −4.44973106395476920719562517510, −4.08886402821955211545716794824, −3.57762111184959094301095677717, −3.36247400375435081295923620288, −2.84501197734112669952025658285, −2.37347746651680436046205368475, −2.23998013252476010545906824429, −1.15002283128956167173797349774, −1.10098203500113952707320934138, 0, 0,
1.10098203500113952707320934138, 1.15002283128956167173797349774, 2.23998013252476010545906824429, 2.37347746651680436046205368475, 2.84501197734112669952025658285, 3.36247400375435081295923620288, 3.57762111184959094301095677717, 4.08886402821955211545716794824, 4.44973106395476920719562517510, 4.45234389988403752872465081444, 5.17621653249316015149339453638, 5.45039613103266583301269945839, 6.01365968308380022181433890173, 6.11454383503353829474283256633, 6.77605196387338580151876308122, 6.79376092553121232131899446021, 7.22428003356006791214523657559, 7.29576348138336129242005600587
