| L(s) = 1 | + 14·3-s − 2.73e3·5-s − 4.80e3·7-s + 1.84e4·9-s − 4.49e4·11-s + 1.00e5·13-s − 3.82e4·15-s − 8.70e5·17-s − 5.08e5·19-s − 6.72e4·21-s − 7.98e4·23-s + 2.69e6·25-s + 7.88e5·27-s + 2.00e6·29-s − 2.18e6·31-s − 6.29e5·33-s + 1.31e7·35-s − 2.07e7·37-s + 1.40e6·39-s + 1.90e7·41-s − 4.19e6·43-s − 5.02e7·45-s + 7.45e7·47-s + 1.72e7·49-s − 1.21e7·51-s − 3.23e6·53-s + 1.22e8·55-s + ⋯ |
| L(s) = 1 | + 0.0997·3-s − 1.95·5-s − 0.755·7-s + 0.935·9-s − 0.925·11-s + 0.973·13-s − 0.194·15-s − 2.52·17-s − 0.895·19-s − 0.0754·21-s − 0.0594·23-s + 1.38·25-s + 0.285·27-s + 0.526·29-s − 0.425·31-s − 0.0923·33-s + 1.47·35-s − 1.81·37-s + 0.0971·39-s + 1.05·41-s − 0.187·43-s − 1.82·45-s + 2.22·47-s + 3/7·49-s − 0.252·51-s − 0.0563·53-s + 1.80·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.1500784444\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1500784444\) |
| \(L(\frac{11}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p^{4} T )^{2} \) |
| 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
−11.97522083621069047047332408526, −11.70821549297374382700664845082, −10.81869814273198786265970996829, −10.78357845134888579896019843922, −10.23143695870121057083905311385, −9.280408522815126807914030430779, −8.585397794619714628596381828510, −8.578989212147553950771210624452, −7.71472610467461015050276229313, −7.14594433182344877090832915868, −6.81408685502384199533394343811, −6.16318604668279791165474188114, −5.20699523842490408126911721651, −4.33445061899249818514886315591, −4.05724606289318206373335108054, −3.65162279158061400724158087090, −2.64798053086491042368075119889, −2.06474294266778208537135673935, −0.906705503307740010653591920549, −0.12250774552239681175279965197,
0.12250774552239681175279965197, 0.906705503307740010653591920549, 2.06474294266778208537135673935, 2.64798053086491042368075119889, 3.65162279158061400724158087090, 4.05724606289318206373335108054, 4.33445061899249818514886315591, 5.20699523842490408126911721651, 6.16318604668279791165474188114, 6.81408685502384199533394343811, 7.14594433182344877090832915868, 7.71472610467461015050276229313, 8.578989212147553950771210624452, 8.585397794619714628596381828510, 9.280408522815126807914030430779, 10.23143695870121057083905311385, 10.78357845134888579896019843922, 10.81869814273198786265970996829, 11.70821549297374382700664845082, 11.97522083621069047047332408526
