| L(s) = 1 | + 2-s − 2·4-s − 4·5-s + 7-s − 3·8-s − 4·10-s − 2·11-s − 13-s + 14-s + 16-s − 6·17-s − 11·19-s + 8·20-s − 2·22-s + 2·23-s + 7·25-s − 26-s − 2·28-s − 10·29-s + 2·32-s − 6·34-s − 4·35-s + 3·37-s − 11·38-s + 12·40-s − 5·41-s − 9·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 4-s − 1.78·5-s + 0.377·7-s − 1.06·8-s − 1.26·10-s − 0.603·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 2.52·19-s + 1.78·20-s − 0.426·22-s + 0.417·23-s + 7/5·25-s − 0.196·26-s − 0.377·28-s − 1.85·29-s + 0.353·32-s − 1.02·34-s − 0.676·35-s + 0.493·37-s − 1.78·38-s + 1.89·40-s − 0.780·41-s − 1.37·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363609 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363609 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 67 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 4 T + 9 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 13 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + T + 25 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 11 T + 67 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 63 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 3 T + 75 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 5 T + 57 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 9 T + 75 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 7 T + 105 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 61 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 9 T + 131 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 103 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 11 T + 127 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 13 T + 197 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 16 T + 197 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 149 T^{2} + p^{2} T^{4} \) |
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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
−10.81214657529496654597769497112, −10.00959936159033348597776320349, −9.464594110082577733076258448341, −9.016048994271286490306679066449, −8.445731196335719348114275801095, −8.420139651698023726496405213253, −7.88628532846598168900550746630, −7.42614425224055220473087444425, −6.64036157316478433719454612144, −6.62620863835549729806059986985, −5.61072706132316862675288028302, −5.18537169999829008006928119132, −4.54036594681705717269483140546, −4.43121347836597456731472352649, −3.86249136899244299642754213368, −3.57006435460292128614229505202, −2.61482237613182721219239131150, −1.89061907082476285138217470449, 0, 0,
1.89061907082476285138217470449, 2.61482237613182721219239131150, 3.57006435460292128614229505202, 3.86249136899244299642754213368, 4.43121347836597456731472352649, 4.54036594681705717269483140546, 5.18537169999829008006928119132, 5.61072706132316862675288028302, 6.62620863835549729806059986985, 6.64036157316478433719454612144, 7.42614425224055220473087444425, 7.88628532846598168900550746630, 8.420139651698023726496405213253, 8.445731196335719348114275801095, 9.016048994271286490306679066449, 9.464594110082577733076258448341, 10.00959936159033348597776320349, 10.81214657529496654597769497112
