| L(s) = 1 | + 488·7-s − 5.45e3·13-s + 1.07e4·19-s + 2.72e4·25-s − 2.03e4·31-s + 1.30e5·37-s + 9.89e4·43-s − 5.66e4·49-s + 2.01e5·61-s + 8.71e5·67-s + 1.23e6·73-s − 1.02e6·79-s − 2.66e6·91-s + 8.54e4·97-s + 2.84e6·103-s + 2.80e5·109-s − 2.18e6·121-s + 127-s + 131-s + 5.26e6·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 1.42·7-s − 2.48·13-s + 1.57·19-s + 1.74·25-s − 0.682·31-s + 2.56·37-s + 1.24·43-s − 0.481·49-s + 0.886·61-s + 2.89·67-s + 3.18·73-s − 2.08·79-s − 3.53·91-s + 0.0935·97-s + 2.60·103-s + 0.216·109-s − 1.23·121-s + 2.23·133-s + 2.62·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(3.398796423\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.398796423\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 1088 p^{2} T^{2} + p^{12} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 244 T + p^{6} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 2182606 T^{2} + p^{12} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2728 T + p^{6} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 13547360 T^{2} + p^{12} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 5392 T + p^{6} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 289848386 T^{2} + p^{12} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 575602720 T^{2} + p^{12} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 10172 T + p^{6} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 65006 T + p^{6} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 4637059040 T^{2} + p^{12} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 49480 T + p^{6} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 5600983390 T^{2} + p^{12} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 43709554208 T^{2} + p^{12} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 44614042990 T^{2} + p^{12} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 100610 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 435736 T + p^{6} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 252585347330 T^{2} + p^{12} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 619568 T + p^{6} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 514340 T + p^{6} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 594384805586 T^{2} + p^{12} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 810559043264 T^{2} + p^{12} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 42704 T + p^{6} T^{2} )^{2} \) |
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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
−12.23909437093049553625119935263, −11.59699758446124471075238294406, −11.23940373367742370849916580287, −10.94694450333794800388663671687, −9.860730350973388316365323358594, −9.860020745158811528311007197193, −9.220340585347905032672667919505, −8.526075741908473970468927427051, −7.70569731628631915516692443021, −7.69057638228113569459024644576, −7.06091044702677641275808680522, −6.33103942886111571040949466708, −5.23664017345010824141485313057, −5.10283548030469841643446138010, −4.63008835440417881123834252963, −3.71448859292368612564012358881, −2.61015361811554698029106023033, −2.36933422790825144710130521220, −1.23088329867627156925572204342, −0.61897037394923263601546122252,
0.61897037394923263601546122252, 1.23088329867627156925572204342, 2.36933422790825144710130521220, 2.61015361811554698029106023033, 3.71448859292368612564012358881, 4.63008835440417881123834252963, 5.10283548030469841643446138010, 5.23664017345010824141485313057, 6.33103942886111571040949466708, 7.06091044702677641275808680522, 7.69057638228113569459024644576, 7.70569731628631915516692443021, 8.526075741908473970468927427051, 9.220340585347905032672667919505, 9.860020745158811528311007197193, 9.860730350973388316365323358594, 10.94694450333794800388663671687, 11.23940373367742370849916580287, 11.59699758446124471075238294406, 12.23909437093049553625119935263
