| L(s) = 1 | + 32·2-s + 14·3-s + 768·4-s + 448·6-s − 4.80e3·7-s + 1.63e4·8-s + 1.84e4·9-s + 4.49e4·11-s + 1.07e4·12-s − 1.00e5·13-s − 1.53e5·14-s + 3.27e5·16-s + 8.70e5·17-s + 5.88e5·18-s + 5.08e5·19-s − 6.72e4·21-s + 1.43e6·22-s − 7.98e4·23-s + 2.29e5·24-s − 3.20e6·26-s + 7.88e5·27-s − 3.68e6·28-s + 2.00e6·29-s + 2.18e6·31-s + 6.29e6·32-s + 6.29e5·33-s + 2.78e7·34-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.0997·3-s + 3/2·4-s + 0.141·6-s − 0.755·7-s + 1.41·8-s + 0.935·9-s + 0.925·11-s + 0.149·12-s − 0.973·13-s − 1.06·14-s + 5/4·16-s + 2.52·17-s + 1.32·18-s + 0.895·19-s − 0.0754·21-s + 1.30·22-s − 0.0594·23-s + 0.141·24-s − 1.37·26-s + 0.285·27-s − 1.13·28-s + 0.526·29-s + 0.425·31-s + 1.06·32-s + 0.0923·33-s + 3.57·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(18.25241325\) |
| \(L(\frac12)\) |
\(\approx\) |
\(18.25241325\) |
| \(L(\frac{11}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{4} T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p^{4} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 14 T - 6070 p T^{2} - 14 p^{9} T^{3} + p^{18} T^{4} \) |
| 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
−10.08629964936321696418959539759, −9.891036051090739838553474405787, −9.433660205296389057600132899384, −8.958816494811093919316293634469, −7.972107374444127906329010421625, −7.58378311878354318635837107701, −7.36851827064527936830845027545, −6.82255569200237279323831515884, −6.19089457216132332563497521224, −5.87969234190704685678583377141, −5.37601997375502343448723108364, −4.79476656166497799385138610315, −4.24072936494116736125912292482, −3.88119375288719906035970377190, −3.29529338976208658798338657403, −2.83267571465240037716271992142, −2.41976651157186546984911692236, −1.49962632664499113868528433657, −1.00773499789087693251589739433, −0.70325250073492125303605733855,
0.70325250073492125303605733855, 1.00773499789087693251589739433, 1.49962632664499113868528433657, 2.41976651157186546984911692236, 2.83267571465240037716271992142, 3.29529338976208658798338657403, 3.88119375288719906035970377190, 4.24072936494116736125912292482, 4.79476656166497799385138610315, 5.37601997375502343448723108364, 5.87969234190704685678583377141, 6.19089457216132332563497521224, 6.82255569200237279323831515884, 7.36851827064527936830845027545, 7.58378311878354318635837107701, 7.972107374444127906329010421625, 8.958816494811093919316293634469, 9.433660205296389057600132899384, 9.891036051090739838553474405787, 10.08629964936321696418959539759
