| L(s) = 1 | + 2-s + 2·3-s − 2·4-s + 2·6-s + 4·7-s − 3·8-s + 3·9-s − 6·11-s − 4·12-s + 2·13-s + 4·14-s + 16-s − 2·17-s + 3·18-s − 8·19-s + 8·21-s − 6·22-s + 4·23-s − 6·24-s + 2·26-s + 4·27-s − 8·28-s − 2·29-s − 12·31-s + 2·32-s − 12·33-s − 2·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s − 4-s + 0.816·6-s + 1.51·7-s − 1.06·8-s + 9-s − 1.80·11-s − 1.15·12-s + 0.554·13-s + 1.06·14-s + 1/4·16-s − 0.485·17-s + 0.707·18-s − 1.83·19-s + 1.74·21-s − 1.27·22-s + 0.834·23-s − 1.22·24-s + 0.392·26-s + 0.769·27-s − 1.51·28-s − 0.371·29-s − 2.15·31-s + 0.353·32-s − 2.08·33-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 107 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_4$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 49 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 54 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 90 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 98 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 14 T + 150 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 10 T + 154 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 153 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 182 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 10 T + 198 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 150 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−7.82490581363402609895073930120, −7.34954707772589817417225807981, −7.18195783210622157068729718330, −6.67941045769765296863226756595, −6.03400845217464075315126936096, −5.93214137300887248246665424039, −5.30541521328240020489861122709, −5.07150435215031521292060590305, −4.70059254549793649310868406102, −4.66567364223158795447316061476, −3.97331145839338364719124844134, −3.94508790956524740744544611983, −3.28531385265656647994357362455, −3.07306552296843957228983013098, −2.39501068571879020484314113277, −2.14224589068375308782464962675, −1.68640141141037256865073922142, −1.23681592256925722731025980959, 0, 0,
1.23681592256925722731025980959, 1.68640141141037256865073922142, 2.14224589068375308782464962675, 2.39501068571879020484314113277, 3.07306552296843957228983013098, 3.28531385265656647994357362455, 3.94508790956524740744544611983, 3.97331145839338364719124844134, 4.66567364223158795447316061476, 4.70059254549793649310868406102, 5.07150435215031521292060590305, 5.30541521328240020489861122709, 5.93214137300887248246665424039, 6.03400845217464075315126936096, 6.67941045769765296863226756595, 7.18195783210622157068729718330, 7.34954707772589817417225807981, 7.82490581363402609895073930120
