| L(s) = 1 | − 2·2-s + 3-s − 4-s + 2·5-s − 2·6-s + 2·7-s + 8·8-s − 4·9-s − 4·10-s − 12-s − 7·13-s − 4·14-s + 2·15-s − 7·16-s + 7·17-s + 8·18-s − 6·19-s − 2·20-s + 2·21-s + 8·24-s + 3·25-s + 14·26-s − 6·27-s − 2·28-s + 3·29-s − 4·30-s − 2·31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s − 1/2·4-s + 0.894·5-s − 0.816·6-s + 0.755·7-s + 2.82·8-s − 4/3·9-s − 1.26·10-s − 0.288·12-s − 1.94·13-s − 1.06·14-s + 0.516·15-s − 7/4·16-s + 1.69·17-s + 1.88·18-s − 1.37·19-s − 0.447·20-s + 0.436·21-s + 1.63·24-s + 3/5·25-s + 2.74·26-s − 1.15·27-s − 0.377·28-s + 0.557·29-s − 0.730·30-s − 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17935225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17935225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 3 | $D_{4}$ | \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 7 T + 37 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 7 T + 45 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 42 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T + p T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 70 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 82 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 7 T + 95 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 20 T + 186 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 18 T + 194 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 14 T + 166 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 98 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - T + 41 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 9 T + 65 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 21 T + 257 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 7 T + 177 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 214 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 9 T + 213 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
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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
−8.193333173722537606100795325114, −8.062784574604308547327946844031, −7.72087665130149865843023055184, −7.53531295067466375426325369711, −6.79355401151260609344822584873, −6.65297789371759523594469150482, −5.76700250181760290931305845211, −5.70540033343584314175556872129, −5.05380734007643271785858437511, −5.00670580174193105860714423774, −4.52807572903839106160818386938, −4.17613814742476707529600833221, −3.28825003933369515932946169660, −3.25442783210435847192640650445, −2.28985737067556298536090522982, −2.27096902317978875114094501300, −1.49670755904099330445475592427, −1.14638474255348899417740599639, 0, 0,
1.14638474255348899417740599639, 1.49670755904099330445475592427, 2.27096902317978875114094501300, 2.28985737067556298536090522982, 3.25442783210435847192640650445, 3.28825003933369515932946169660, 4.17613814742476707529600833221, 4.52807572903839106160818386938, 5.00670580174193105860714423774, 5.05380734007643271785858437511, 5.70540033343584314175556872129, 5.76700250181760290931305845211, 6.65297789371759523594469150482, 6.79355401151260609344822584873, 7.53531295067466375426325369711, 7.72087665130149865843023055184, 8.062784574604308547327946844031, 8.193333173722537606100795325114
