| L(s) = 1 | − 32·2-s + 14·3-s + 768·4-s + 2.73e3·5-s − 448·6-s − 1.63e4·8-s + 1.84e4·9-s − 8.73e4·10-s + 4.49e4·11-s + 1.07e4·12-s − 1.00e5·13-s + 3.82e4·15-s + 3.27e5·16-s + 8.70e5·17-s − 5.88e5·18-s − 5.08e5·19-s + 2.09e6·20-s − 1.43e6·22-s + 7.98e4·23-s − 2.29e5·24-s + 2.69e6·25-s + 3.20e6·26-s + 7.88e5·27-s + 2.00e6·29-s − 1.22e6·30-s − 2.18e6·31-s − 6.29e6·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.0997·3-s + 3/2·4-s + 1.95·5-s − 0.141·6-s − 1.41·8-s + 0.935·9-s − 2.76·10-s + 0.925·11-s + 0.149·12-s − 0.973·13-s + 0.194·15-s + 5/4·16-s + 2.52·17-s − 1.32·18-s − 0.895·19-s + 2.93·20-s − 1.30·22-s + 0.0594·23-s − 0.141·24-s + 1.38·25-s + 1.37·26-s + 0.285·27-s + 0.526·29-s − 0.275·30-s − 0.425·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.129475335\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.129475335\) |
| \(L(\frac{11}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{4} T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 14 T - 6070 p T^{2} - 14 p^{9} T^{3} + p^{18} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 546 p T + 950594 p T^{2} - 546 p^{10} T^{3} + p^{18} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 44940 T + 2679533782 T^{2} - 44940 p^{9} T^{3} + p^{18} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 7714 p T + 23606428002 T^{2} + 7714 p^{10} T^{3} + p^{18} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 870408 T + 420751004110 T^{2} - 870408 p^{9} T^{3} + p^{18} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 508774 T + 15410410458 p T^{2} + 508774 p^{9} T^{3} + p^{18} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 79800 T + 3486389354926 T^{2} - 79800 p^{9} T^{3} + p^{18} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2006328 T - 2889837567866 T^{2} - 2006328 p^{9} T^{3} + p^{18} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2188732 T + 50511394548798 T^{2} + 2188732 p^{9} T^{3} + p^{18} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 20723576 T + 320148592616598 T^{2} + 20723576 p^{9} T^{3} + p^{18} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 19016592 T + 126947391521038 T^{2} + 19016592 p^{9} T^{3} + p^{18} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4193716 T + 733843976191350 T^{2} - 4193716 p^{9} T^{3} + p^{18} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 74542524 T + 3534811031083678 T^{2} - 74542524 p^{9} T^{3} + p^{18} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3239748 T + 6104780971308142 T^{2} + 3239748 p^{9} T^{3} + p^{18} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 133642362 T + 18901037953676014 T^{2} - 133642362 p^{9} T^{3} + p^{18} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 227801686 T + 34632227131611306 T^{2} + 227801686 p^{9} T^{3} + p^{18} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 332930272 T + 75814987106046390 T^{2} - 332930272 p^{9} T^{3} + p^{18} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 167985720 T + 6569741497979662 T^{2} + 167985720 p^{9} T^{3} + p^{18} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 44684276 T + 34064802404568870 T^{2} - 44684276 p^{9} T^{3} + p^{18} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 269642776 T + 246583195950001182 T^{2} - 269642776 p^{9} T^{3} + p^{18} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 183105762 T + 297182791992067342 T^{2} - 183105762 p^{9} T^{3} + p^{18} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 791657748 T + 751278555124390294 T^{2} + 791657748 p^{9} T^{3} + p^{18} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4169480 T + 1069351625837487534 T^{2} - 4169480 p^{9} T^{3} + p^{18} T^{4} \) |
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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
−12.35829094245976706339592985832, −11.95621801494012932014941899150, −11.00529087954350185761522192926, −10.18583586344390301684429741488, −10.12644909036089991879773064990, −9.910197857143110883629239469613, −9.087745511484869929114577609298, −8.923751966904666789746702288290, −7.984031453049583935924865975154, −7.41822321796819488523426894992, −6.83116626036244688420277619934, −6.36981301071607788447844058256, −5.56960918888171157739784238834, −5.25917269491980614304805987692, −4.02174750087763800153197190446, −3.14497718489080639100111833564, −2.29444438397299505807549402940, −1.72334655430287513571037709543, −1.32848085843790126318020086955, −0.61081025555080911694194492422,
0.61081025555080911694194492422, 1.32848085843790126318020086955, 1.72334655430287513571037709543, 2.29444438397299505807549402940, 3.14497718489080639100111833564, 4.02174750087763800153197190446, 5.25917269491980614304805987692, 5.56960918888171157739784238834, 6.36981301071607788447844058256, 6.83116626036244688420277619934, 7.41822321796819488523426894992, 7.984031453049583935924865975154, 8.923751966904666789746702288290, 9.087745511484869929114577609298, 9.910197857143110883629239469613, 10.12644909036089991879773064990, 10.18583586344390301684429741488, 11.00529087954350185761522192926, 11.95621801494012932014941899150, 12.35829094245976706339592985832
