| L(s) = 1 | − 2·3-s − 5-s − 3·7-s + 3·9-s + 2·11-s − 2·13-s + 2·15-s + 8·17-s − 12·19-s + 6·21-s + 7·23-s + 25-s − 4·27-s − 7·29-s + 2·31-s − 4·33-s + 3·35-s + 2·37-s + 4·39-s − 7·41-s + 43-s − 3·45-s − 8·47-s + 3·49-s − 16·51-s + 2·53-s − 2·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s − 1.13·7-s + 9-s + 0.603·11-s − 0.554·13-s + 0.516·15-s + 1.94·17-s − 2.75·19-s + 1.30·21-s + 1.45·23-s + 1/5·25-s − 0.769·27-s − 1.29·29-s + 0.359·31-s − 0.696·33-s + 0.507·35-s + 0.328·37-s + 0.640·39-s − 1.09·41-s + 0.152·43-s − 0.447·45-s − 1.16·47-s + 3/7·49-s − 2.24·51-s + 0.274·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47114496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47114496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 6 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 7 T + 48 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 7 T + 60 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 7 T + 84 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - T - 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 66 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 21 T + 218 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 15 T + 168 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 9 T + 62 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 10 T + 126 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 3 T + 138 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 126 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 18 T + 206 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 10 T + 162 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−7.61001819386563567709428677532, −7.24612019585631400747452020544, −7.02969208184865406969157776201, −6.79344739535017758290939564891, −6.31277222700592125955929369751, −6.12413632941585373974236427854, −5.74804845492896377906033119683, −5.32120795772886066610674338942, −4.94086599898997524774157619357, −4.72818415603061453462062339144, −3.98431247534232072523416869895, −3.91986791503091276404618519177, −3.48406217765785004410411241504, −3.11246846985585081597159256473, −2.32303176223531113228317210951, −2.22459892140817555067716360000, −1.22362002679157335275323606117, −1.04097139636600713320524204073, 0, 0,
1.04097139636600713320524204073, 1.22362002679157335275323606117, 2.22459892140817555067716360000, 2.32303176223531113228317210951, 3.11246846985585081597159256473, 3.48406217765785004410411241504, 3.91986791503091276404618519177, 3.98431247534232072523416869895, 4.72818415603061453462062339144, 4.94086599898997524774157619357, 5.32120795772886066610674338942, 5.74804845492896377906033119683, 6.12413632941585373974236427854, 6.31277222700592125955929369751, 6.79344739535017758290939564891, 7.02969208184865406969157776201, 7.24612019585631400747452020544, 7.61001819386563567709428677532
