| L(s) = 1 | + 2·2-s + 3·4-s − 3·5-s + 4·8-s − 6·10-s − 6·11-s − 2·13-s + 5·16-s + 6·17-s + 7·19-s − 9·20-s − 12·22-s + 3·23-s + 5·25-s − 4·26-s + 6·29-s + 4·31-s + 6·32-s + 12·34-s − 2·37-s + 14·38-s − 12·40-s − 2·43-s − 18·44-s + 6·46-s + 10·50-s − 6·52-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 1.34·5-s + 1.41·8-s − 1.89·10-s − 1.80·11-s − 0.554·13-s + 5/4·16-s + 1.45·17-s + 1.60·19-s − 2.01·20-s − 2.55·22-s + 0.625·23-s + 25-s − 0.784·26-s + 1.11·29-s + 0.718·31-s + 1.06·32-s + 2.05·34-s − 0.328·37-s + 2.27·38-s − 1.89·40-s − 0.304·43-s − 2.71·44-s + 0.884·46-s + 1.41·50-s − 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.049478911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.049478911\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−8.858950556923864377819119323278, −8.482083733865472863739072583068, −8.042890366432348679201748384642, −7.81483241923227660534897359869, −7.38142009381650076942374024511, −7.34813278557772562998255303352, −6.74491255579741140628666536701, −6.37983515279038426289165636690, −5.66977308422121014614917719817, −5.45803972849861360057363719879, −4.97916902072445454831474191262, −4.90697133009652670068329192723, −4.41042664073731870777256956837, −3.69505231734168906113720480524, −3.41390121704826515959585786245, −3.15387774915306592228537812419, −2.56211467694364357154000062474, −2.27440605358588165145481548380, −1.14394034643935806918428788045, −0.65171701235582803956063941108,
0.65171701235582803956063941108, 1.14394034643935806918428788045, 2.27440605358588165145481548380, 2.56211467694364357154000062474, 3.15387774915306592228537812419, 3.41390121704826515959585786245, 3.69505231734168906113720480524, 4.41042664073731870777256956837, 4.90697133009652670068329192723, 4.97916902072445454831474191262, 5.45803972849861360057363719879, 5.66977308422121014614917719817, 6.37983515279038426289165636690, 6.74491255579741140628666536701, 7.34813278557772562998255303352, 7.38142009381650076942374024511, 7.81483241923227660534897359869, 8.042890366432348679201748384642, 8.482083733865472863739072583068, 8.858950556923864377819119323278
