| L(s) = 1 | − 2-s + 2·3-s − 2·4-s − 2·6-s − 7-s + 3·8-s − 3·9-s − 6·11-s − 4·12-s − 3·13-s + 14-s + 16-s − 6·17-s + 3·18-s − 5·19-s − 2·21-s + 6·22-s + 12·23-s + 6·24-s + 3·26-s − 14·27-s + 2·28-s − 5·29-s − 6·31-s − 2·32-s − 12·33-s + 6·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s − 4-s − 0.816·6-s − 0.377·7-s + 1.06·8-s − 9-s − 1.80·11-s − 1.15·12-s − 0.832·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.707·18-s − 1.14·19-s − 0.436·21-s + 1.27·22-s + 2.50·23-s + 1.22·24-s + 0.588·26-s − 2.69·27-s + 0.377·28-s − 0.928·29-s − 1.07·31-s − 0.353·32-s − 2.08·33-s + 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 + T + 13 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 17 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 5 T + 33 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 63 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 73 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 86 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3 T + 77 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + T + 93 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 87 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 15 T + 163 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 81 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 14 T + 178 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 11 T + 171 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 5 T + 133 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 177 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + T + 183 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−10.01652423904056671200617946383, −9.970834657430971716653455921207, −9.187698812468083088655451654991, −9.101642758873869954633728325187, −8.637477577960095460581849441720, −8.506646481105811914530586321941, −7.88479164562301327349912281171, −7.60350053975144588214021745216, −7.01893864734912219439462109009, −6.49202348976061418970324222979, −5.60320073730555610124605744204, −5.40322716195685121609769276044, −4.58034193435069674182828980796, −4.55342045471447278041966163734, −3.37925279640691363572939029312, −3.07879746442872269965969755136, −2.51247516856397435050198902781, −1.92322418237009403501123881674, 0, 0,
1.92322418237009403501123881674, 2.51247516856397435050198902781, 3.07879746442872269965969755136, 3.37925279640691363572939029312, 4.55342045471447278041966163734, 4.58034193435069674182828980796, 5.40322716195685121609769276044, 5.60320073730555610124605744204, 6.49202348976061418970324222979, 7.01893864734912219439462109009, 7.60350053975144588214021745216, 7.88479164562301327349912281171, 8.506646481105811914530586321941, 8.637477577960095460581849441720, 9.101642758873869954633728325187, 9.187698812468083088655451654991, 9.970834657430971716653455921207, 10.01652423904056671200617946383
