| L(s) = 1 | − 8·2-s + 54·3-s − 84·4-s − 432·6-s − 86·7-s + 832·8-s + 2.18e3·9-s − 4.61e3·11-s − 4.53e3·12-s − 9.01e3·13-s + 688·14-s − 2.41e3·16-s + 1.99e4·17-s − 1.74e4·18-s − 2.39e4·19-s − 4.64e3·21-s + 3.68e4·22-s − 1.98e4·23-s + 4.49e4·24-s + 7.21e4·26-s + 7.87e4·27-s + 7.22e3·28-s − 3.46e5·29-s − 1.29e5·31-s + 1.13e5·32-s − 2.49e5·33-s − 1.59e5·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s − 0.656·4-s − 0.816·6-s − 0.0947·7-s + 0.574·8-s + 9-s − 1.04·11-s − 0.757·12-s − 1.13·13-s + 0.0670·14-s − 0.147·16-s + 0.984·17-s − 0.707·18-s − 0.800·19-s − 0.109·21-s + 0.738·22-s − 0.339·23-s + 0.663·24-s + 0.804·26-s + 0.769·27-s + 0.0621·28-s − 2.63·29-s − 0.778·31-s + 0.611·32-s − 1.20·33-s − 0.696·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| 5 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + p^{3} T + 37 p^{2} T^{2} + p^{10} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 86 T + 1624631 T^{2} + 86 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4612 T + 35518234 T^{2} + 4612 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 9018 T + 134963731 T^{2} + 9018 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 19948 T + 314529622 T^{2} - 19948 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 23934 T + 995928583 T^{2} + 23934 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 19812 T + 5039109154 T^{2} + 19812 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 346508 T + 64488369118 T^{2} + 346508 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 129178 T + 29398278647 T^{2} + 129178 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 58036 T - 15345459714 T^{2} + 58036 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 340928 T + 203171618242 T^{2} + 340928 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 1087350 T + 829144455535 T^{2} + 1087350 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 448420 T + 1032529649890 T^{2} - 448420 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 728552 T + 795170826394 T^{2} + 728552 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 907844 T + 3501429103018 T^{2} - 907844 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 478410 T - 2138250815357 T^{2} + 478410 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4514702 T + 17215857407447 T^{2} + 4514702 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 2607296 T + 17905773132286 T^{2} + 2607296 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4070268 T + 25823849945014 T^{2} - 4070268 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6227360 T + 47990625121118 T^{2} - 6227360 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 5769084 T + 41393756775802 T^{2} - 5769084 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 16356096 T + 133666580484178 T^{2} - 16356096 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 17433442 T + 204531330485667 T^{2} + 17433442 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−12.84941829857426625178435627891, −12.55555259579185137154124662171, −11.74476451547060353048354266782, −10.96815576616069821088150632545, −10.13674727801174657776992274864, −9.995768414766857779337021562429, −9.190932805426060003574524924154, −9.077702120540965566695246899454, −8.028500625939717141575204089163, −7.952878563514913985262948339296, −7.31565737839715962067225901613, −6.43225729300978571403629594816, −5.28527509469159864412530517113, −4.88479577188490182659343609434, −3.84313931145968140498812204120, −3.23518063942483111155122389749, −2.26290673483007228371179917995, −1.60175371726064610287877131697, 0, 0,
1.60175371726064610287877131697, 2.26290673483007228371179917995, 3.23518063942483111155122389749, 3.84313931145968140498812204120, 4.88479577188490182659343609434, 5.28527509469159864412530517113, 6.43225729300978571403629594816, 7.31565737839715962067225901613, 7.952878563514913985262948339296, 8.028500625939717141575204089163, 9.077702120540965566695246899454, 9.190932805426060003574524924154, 9.995768414766857779337021562429, 10.13674727801174657776992274864, 10.96815576616069821088150632545, 11.74476451547060353048354266782, 12.55555259579185137154124662171, 12.84941829857426625178435627891
