| L(s) = 1 | − 4·2-s + 7·3-s + 12·4-s − 28·6-s + 5·7-s − 32·8-s + 9·9-s − 51·11-s + 84·12-s − 26·13-s − 20·14-s + 80·16-s + 63·17-s − 36·18-s + 28·19-s + 35·21-s + 204·22-s + 9·23-s − 224·24-s + 104·26-s − 28·27-s + 60·28-s − 198·29-s + 175·31-s − 192·32-s − 357·33-s − 252·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.34·3-s + 3/2·4-s − 1.90·6-s + 0.269·7-s − 1.41·8-s + 1/3·9-s − 1.39·11-s + 2.02·12-s − 0.554·13-s − 0.381·14-s + 5/4·16-s + 0.898·17-s − 0.471·18-s + 0.338·19-s + 0.363·21-s + 1.97·22-s + 0.0815·23-s − 1.90·24-s + 0.784·26-s − 0.199·27-s + 0.404·28-s − 1.26·29-s + 1.01·31-s − 1.06·32-s − 1.88·33-s − 1.27·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 422500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 422500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.909535402\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.909535402\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 7 T + 40 T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 5 T + 666 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 51 T + 2656 T^{2} + 51 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 63 T + 9532 T^{2} - 63 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 28 T + 10134 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 9 T + 16768 T^{2} - 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 198 T + 56899 T^{2} + 198 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 175 T + 64062 T^{2} - 175 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 632 T + 197382 T^{2} - 632 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 210 T + 136162 T^{2} - 210 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 23 T + 151560 T^{2} - 23 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 231 T + 129610 T^{2} - 231 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 186 T + 285823 T^{2} + 186 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 63 T + 410464 T^{2} - 63 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 182 T + 240063 T^{2} + 182 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 695 T + 523596 T^{2} - 695 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 390 T - 193778 T^{2} - 390 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1526 T + 1231578 T^{2} - 1526 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 863 T + 991434 T^{2} + 863 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 315 T + 585604 T^{2} - 315 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1638 T + 1580794 T^{2} + 1638 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 98 T + 996042 T^{2} - 98 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.14457539118605069147347330741, −9.668300201491275622154802080968, −9.426879682177825617975149046077, −9.323951962850405555687678826758, −8.344792590739361304099178245460, −8.249028020448431955275621115210, −7.88291219545671808973735537601, −7.72687333271058219638906277977, −7.11070658147207233406910946179, −6.63577798542684304804278507107, −5.72149247477053852037659481783, −5.71167017401484748595428607984, −4.82751247384841894313216212725, −4.27671325650200537056196266278, −3.17863618884491847534005883639, −3.17215838905741677826473202020, −2.29552048444035712697478301742, −2.23989294362442183115129447094, −1.12651675474771976048296918426, −0.49776152233045255375828449596,
0.49776152233045255375828449596, 1.12651675474771976048296918426, 2.23989294362442183115129447094, 2.29552048444035712697478301742, 3.17215838905741677826473202020, 3.17863618884491847534005883639, 4.27671325650200537056196266278, 4.82751247384841894313216212725, 5.71167017401484748595428607984, 5.72149247477053852037659481783, 6.63577798542684304804278507107, 7.11070658147207233406910946179, 7.72687333271058219638906277977, 7.88291219545671808973735537601, 8.249028020448431955275621115210, 8.344792590739361304099178245460, 9.323951962850405555687678826758, 9.426879682177825617975149046077, 9.668300201491275622154802080968, 10.14457539118605069147347330741
