| L(s) = 1 | + 4.48e3·9-s − 2.05e5·11-s + 2.74e5·19-s + 1.37e7·29-s − 5.83e5·31-s + 5.95e7·41-s − 3.12e7·49-s − 1.85e8·59-s + 3.91e8·61-s − 6.22e8·71-s + 1.08e9·79-s + 6.07e8·81-s + 9.24e8·89-s − 9.21e8·99-s − 2.37e9·101-s + 2.90e9·109-s + 1.69e10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.35e10·169-s + ⋯ |
| L(s) = 1 | + 0.227·9-s − 4.23·11-s + 0.483·19-s + 3.61·29-s − 0.113·31-s + 3.29·41-s − 0.773·49-s − 1.99·59-s + 3.61·61-s − 2.90·71-s + 3.13·79-s + 1.56·81-s + 1.56·89-s − 0.963·99-s − 2.26·101-s + 1.97·109-s + 7.20·121-s + 1.27·169-s + 0.110·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(4.765697842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.765697842\) |
| \(L(\frac{11}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 4484 T^{2} - 65252762 p^{2} T^{4} - 4484 p^{18} T^{6} + p^{36} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 31230700 T^{2} + 71158420123302 p^{2} T^{4} + 31230700 p^{18} T^{6} + p^{36} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 102720 T + 7335543382 T^{2} + 102720 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 13506080684 T^{2} + 64066498776709104822 T^{4} - 13506080684 p^{18} T^{6} + p^{36} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 419518771196 T^{2} + \)\(71\!\cdots\!22\)\( T^{4} - 419518771196 p^{18} T^{6} + p^{36} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 137272 T + 111610161654 T^{2} - 137272 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 5330064218900 T^{2} + \)\(13\!\cdots\!38\)\( T^{4} - 5330064218900 p^{18} T^{6} + p^{36} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 6893748 T + 40195999658014 T^{2} - 6893748 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 291832 T + 38964935800398 T^{2} + 291832 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 310941286370060 T^{2} + \)\(48\!\cdots\!58\)\( T^{4} - 310941286370060 p^{18} T^{6} + p^{36} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 29773452 T + 771012402449398 T^{2} - 29773452 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 1540458208886372 T^{2} + \)\(10\!\cdots\!94\)\( T^{4} - 1540458208886372 p^{18} T^{6} + p^{36} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 2450505224578676 T^{2} + \)\(38\!\cdots\!22\)\( T^{4} - 2450505224578676 p^{18} T^{6} + p^{36} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 11327676942925580 T^{2} + \)\(53\!\cdots\!78\)\( T^{4} - 11327676942925580 p^{18} T^{6} + p^{36} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 92930856 T + 16656477955483462 T^{2} + 92930856 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 195673924 T + 22434263296171326 T^{2} - 195673924 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 83008173082523780 T^{2} + \)\(31\!\cdots\!18\)\( T^{4} - 83008173082523780 p^{18} T^{6} + p^{36} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 311207016 T + 76405636625293726 T^{2} + 311207016 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 60959480039527964 T^{2} + \)\(70\!\cdots\!62\)\( T^{4} - 60959480039527964 p^{18} T^{6} + p^{36} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 542261776 T + 313115996157615582 T^{2} - 542261776 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 43663493653967740 T^{2} + \)\(69\!\cdots\!18\)\( T^{4} + 43663493653967740 p^{18} T^{6} + p^{36} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 462291852 T + 159603168035249494 T^{2} - 462291852 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 1302744298917711740 T^{2} + \)\(11\!\cdots\!78\)\( T^{4} - 1302744298917711740 p^{18} T^{6} + p^{36} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.59375804375869854630514932722, −6.32100011129555536287954263001, −6.08509564790683321232505645827, −5.93302501674199644650458251783, −5.56905649102650174424247090693, −5.27258549866603078280545213440, −5.13309297660004196959827738388, −4.95890712187277402752800706302, −4.62978436285858376213584592496, −4.51063149068807991992005308038, −4.30425714794621860756367645641, −3.67378193245519482695237066712, −3.58774079487115715027195336341, −3.04284289830453131673823211521, −3.01833307919507472076585379772, −2.64593925535335767425570499750, −2.49091637931035581941185289538, −2.30766631322051468402236444973, −2.17944952954039954050153134361, −1.61262926726973722118267917690, −1.11660870323694463567053871581, −1.04787415997037746839410776470, −0.53691244650309628237494344902, −0.53457023692732957889662934821, −0.26635940276147155299680305571,
0.26635940276147155299680305571, 0.53457023692732957889662934821, 0.53691244650309628237494344902, 1.04787415997037746839410776470, 1.11660870323694463567053871581, 1.61262926726973722118267917690, 2.17944952954039954050153134361, 2.30766631322051468402236444973, 2.49091637931035581941185289538, 2.64593925535335767425570499750, 3.01833307919507472076585379772, 3.04284289830453131673823211521, 3.58774079487115715027195336341, 3.67378193245519482695237066712, 4.30425714794621860756367645641, 4.51063149068807991992005308038, 4.62978436285858376213584592496, 4.95890712187277402752800706302, 5.13309297660004196959827738388, 5.27258549866603078280545213440, 5.56905649102650174424247090693, 5.93302501674199644650458251783, 6.08509564790683321232505645827, 6.32100011129555536287954263001, 6.59375804375869854630514932722
