| L(s) = 1 | − 3·2-s + 4·4-s + 2·5-s − 2·7-s − 3·8-s − 6·10-s − 4·13-s + 6·14-s + 3·16-s + 3·17-s + 2·19-s + 8·20-s + 7·23-s − 2·25-s + 12·26-s − 8·28-s − 3·29-s − 31-s − 6·32-s − 9·34-s − 4·35-s + 37-s − 6·38-s − 6·40-s − 14·41-s − 3·43-s − 21·46-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 2·4-s + 0.894·5-s − 0.755·7-s − 1.06·8-s − 1.89·10-s − 1.10·13-s + 1.60·14-s + 3/4·16-s + 0.727·17-s + 0.458·19-s + 1.78·20-s + 1.45·23-s − 2/5·25-s + 2.35·26-s − 1.51·28-s − 0.557·29-s − 0.179·31-s − 1.06·32-s − 1.54·34-s − 0.676·35-s + 0.164·37-s − 0.973·38-s − 0.948·40-s − 2.18·41-s − 0.457·43-s − 3.09·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96059601 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96059601 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 25 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 7 T + 57 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 49 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T + 61 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - T + 43 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 14 T + 111 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3 T + 77 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 77 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - T + 117 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 15 T + 167 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 17 T + 183 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 7 T + 147 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - T + 57 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 24 T + 305 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 254 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−7.36564089101001515306373616917, −7.33875246759997590032897136499, −7.01294870627190648388663549634, −6.76407951214699463214181945689, −6.24130496970240728889870074364, −5.71059845801129011258144130997, −5.70717421503230727504299722317, −5.27437373902719871616719709042, −4.73408694094873541013767749480, −4.58084910652470354056953191141, −3.78844922314455836032125852144, −3.31159389581778030894334835561, −3.18461305965359200887251225878, −2.68046754225845279745571171221, −2.15874412037317749922676207128, −1.74890189224612268095813691240, −1.29842138612241608100366981190, −0.903561277900623911536315014866, 0, 0,
0.903561277900623911536315014866, 1.29842138612241608100366981190, 1.74890189224612268095813691240, 2.15874412037317749922676207128, 2.68046754225845279745571171221, 3.18461305965359200887251225878, 3.31159389581778030894334835561, 3.78844922314455836032125852144, 4.58084910652470354056953191141, 4.73408694094873541013767749480, 5.27437373902719871616719709042, 5.70717421503230727504299722317, 5.71059845801129011258144130997, 6.24130496970240728889870074364, 6.76407951214699463214181945689, 7.01294870627190648388663549634, 7.33875246759997590032897136499, 7.36564089101001515306373616917
