| L(s) = 1 | − 2-s − 5·3-s − 11·4-s + 5·6-s + 9·7-s + 15·8-s + 3·9-s + 80·11-s + 55·12-s + 26·13-s − 9·14-s + 61·16-s − 19·17-s − 3·18-s − 84·19-s − 45·21-s − 80·22-s − 196·23-s − 75·24-s − 26·26-s − 40·27-s − 99·28-s − 44·29-s − 86·31-s − 89·32-s − 400·33-s + 19·34-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 0.962·3-s − 1.37·4-s + 0.340·6-s + 0.485·7-s + 0.662·8-s + 1/9·9-s + 2.19·11-s + 1.32·12-s + 0.554·13-s − 0.171·14-s + 0.953·16-s − 0.271·17-s − 0.0392·18-s − 1.01·19-s − 0.467·21-s − 0.775·22-s − 1.77·23-s − 0.637·24-s − 0.196·26-s − 0.285·27-s − 0.668·28-s − 0.281·29-s − 0.498·31-s − 0.491·32-s − 2.11·33-s + 0.0958·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 p^{2} T^{2} + p^{3} T^{3} + p^{6} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + 5 T + 22 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 9 T + 192 T^{2} - 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 80 T + 3650 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 19 T + 8688 T^{2} + 19 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 84 T + 11130 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 196 T + 33326 T^{2} + 196 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 44 T + 10094 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 86 T + 56518 T^{2} + 86 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 209 T + 112120 T^{2} + 209 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 230 T + 149010 T^{2} + 230 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 287 T + 92698 T^{2} + 287 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 435 T + 192728 T^{2} + 435 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 118 T + 297410 T^{2} - 118 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 368 T + 379266 T^{2} + 368 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1058 T + 580378 T^{2} + 1058 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 68 T + 373930 T^{2} + 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 131 T + 493328 T^{2} + 131 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 456 T + 542718 T^{2} + 456 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1008 T + 1233294 T^{2} + 1008 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1958 T + 1961238 T^{2} + 1958 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 720 T + 899726 T^{2} + 720 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 928 T + 943870 T^{2} - 928 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−11.09354610853829987694532161630, −10.33024573802130563589285980905, −10.03622242954707706770117566295, −9.560835643613227700045418389246, −8.854889469359021605773712376934, −8.814353649769759965806317154273, −8.319087508148672862578217346249, −7.72250821500815153686429433420, −6.97935361950376376246739837152, −6.39111661429318702504578573827, −6.07874973425576056185318351006, −5.52112020074594644088136851252, −4.80199635486944999408467554448, −4.28722443161256416007176040759, −3.98588719981268042967142058337, −3.31296382636952452493270107709, −1.63993950518740173165231711389, −1.51355264638722301515686199313, 0, 0,
1.51355264638722301515686199313, 1.63993950518740173165231711389, 3.31296382636952452493270107709, 3.98588719981268042967142058337, 4.28722443161256416007176040759, 4.80199635486944999408467554448, 5.52112020074594644088136851252, 6.07874973425576056185318351006, 6.39111661429318702504578573827, 6.97935361950376376246739837152, 7.72250821500815153686429433420, 8.319087508148672862578217346249, 8.814353649769759965806317154273, 8.854889469359021605773712376934, 9.560835643613227700045418389246, 10.03622242954707706770117566295, 10.33024573802130563589285980905, 11.09354610853829987694532161630
