| L(s) = 1 | − 2·2-s + 3·4-s − 2·5-s + 2·7-s − 4·8-s − 9-s + 4·10-s + 2·13-s − 4·14-s + 5·16-s + 2·17-s + 2·18-s − 6·20-s − 12·23-s + 3·25-s − 4·26-s + 6·28-s − 6·31-s − 6·32-s − 4·34-s − 4·35-s − 3·36-s + 8·40-s − 10·41-s + 2·43-s + 2·45-s + 24·46-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.894·5-s + 0.755·7-s − 1.41·8-s − 1/3·9-s + 1.26·10-s + 0.554·13-s − 1.06·14-s + 5/4·16-s + 0.485·17-s + 0.471·18-s − 1.34·20-s − 2.50·23-s + 3/5·25-s − 0.784·26-s + 1.13·28-s − 1.07·31-s − 1.06·32-s − 0.685·34-s − 0.676·35-s − 1/2·36-s + 1.26·40-s − 1.56·41-s + 0.304·43-s + 0.298·45-s + 3.53·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71740900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71740900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 33 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 10 T + 102 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 82 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 50 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 62 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 77 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_4$ | \( 1 + 16 T + 166 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 10 T + 154 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 158 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 10 T + 166 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 16 T + 217 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 165 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 30 T + 398 T^{2} + 30 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 16 T + 238 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−7.59678279233050086686461187014, −7.42491129862571556621323947336, −7.24700723622687158554385952442, −6.74017309806897451830798333828, −6.15271164089432478869951377365, −6.09974513025417322749803845763, −5.57561026053187447106754248224, −5.47059377605500960068677965096, −4.66475354902551757530558363009, −4.41164144425328976219731921180, −4.09190823856226042749472426122, −3.54391758655861140139800648002, −3.15105347525637893361597702738, −2.96948523505928727454648456026, −2.06557513668394933611581132643, −1.92814177470901809841386216252, −1.47945003216174396944052996806, −0.919957714013372119031431770166, 0, 0,
0.919957714013372119031431770166, 1.47945003216174396944052996806, 1.92814177470901809841386216252, 2.06557513668394933611581132643, 2.96948523505928727454648456026, 3.15105347525637893361597702738, 3.54391758655861140139800648002, 4.09190823856226042749472426122, 4.41164144425328976219731921180, 4.66475354902551757530558363009, 5.47059377605500960068677965096, 5.57561026053187447106754248224, 6.09974513025417322749803845763, 6.15271164089432478869951377365, 6.74017309806897451830798333828, 7.24700723622687158554385952442, 7.42491129862571556621323947336, 7.59678279233050086686461187014
