| L(s) = 1 | + 28·5-s − 54·9-s − 28·17-s + 338·25-s − 1.51e3·45-s + 610·49-s + 1.26e3·61-s − 2.15e3·73-s + 2.18e3·81-s − 784·85-s + 4.06e3·101-s − 1.06e3·121-s + 476·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 1.51e3·153-s + 157-s + 163-s + 167-s + 4.39e3·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 2.50·5-s − 2·9-s − 0.399·17-s + 2.70·25-s − 5.00·45-s + 1.77·49-s + 2.64·61-s − 3.45·73-s + 3·81-s − 1.00·85-s + 3.99·101-s − 0.797·121-s + 0.340·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.798·153-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.131420797\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.131420797\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + p^{3} T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 14 T + p^{3} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 36 T + p^{3} T^{2} )( 1 + 36 T + p^{3} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 40 T + p^{3} T^{2} )( 1 + 40 T + p^{3} T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 14 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 212 T + p^{3} T^{2} )( 1 + 212 T + p^{3} T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 128 T + p^{3} T^{2} )( 1 + 128 T + p^{3} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 364 T + p^{3} T^{2} )( 1 + 364 T + p^{3} T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 630 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 1078 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 112 T + p^{3} T^{2} )( 1 + 112 T + p^{3} T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
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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
−11.59109951650437025559459432859, −11.01362858583989714586401402144, −10.37047619594120002201234304280, −10.18929341130741727878963633339, −9.674650817389476664491236308845, −9.043577731948218315153676159043, −8.872682845806345648756940455059, −8.476361883552909002964419638903, −7.70257924745289055830810014220, −6.99769514987501524780091716335, −6.39456280831263434494664124340, −5.92090758564291346110606477426, −5.66593458287866940354291488654, −5.32020098150390686028939193206, −4.55363134504054964644300450590, −3.54455020040670411745279151553, −2.72872994478072968287361915285, −2.34850825598304071896361425387, −1.75631633602103052798465545814, −0.62897296638198311545909883861,
0.62897296638198311545909883861, 1.75631633602103052798465545814, 2.34850825598304071896361425387, 2.72872994478072968287361915285, 3.54455020040670411745279151553, 4.55363134504054964644300450590, 5.32020098150390686028939193206, 5.66593458287866940354291488654, 5.92090758564291346110606477426, 6.39456280831263434494664124340, 6.99769514987501524780091716335, 7.70257924745289055830810014220, 8.476361883552909002964419638903, 8.872682845806345648756940455059, 9.043577731948218315153676159043, 9.674650817389476664491236308845, 10.18929341130741727878963633339, 10.37047619594120002201234304280, 11.01362858583989714586401402144, 11.59109951650437025559459432859
