| L(s) = 1 | + 16·7-s + 6·9-s − 32·23-s + 40·29-s − 88·41-s − 160·43-s + 224·47-s + 60·49-s − 56·61-s + 96·63-s − 128·67-s + 27·81-s − 224·83-s + 312·89-s + 664·101-s + 16·103-s − 192·107-s + 24·109-s + 260·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 512·161-s + ⋯ |
| L(s) = 1 | + 16/7·7-s + 2/3·9-s − 1.39·23-s + 1.37·29-s − 2.14·41-s − 3.72·43-s + 4.76·47-s + 1.22·49-s − 0.918·61-s + 1.52·63-s − 1.91·67-s + 1/3·81-s − 2.69·83-s + 3.50·89-s + 6.57·101-s + 0.155·103-s − 1.79·107-s + 0.220·109-s + 2.14·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s − 3.18·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(7.885200174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.885200174\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( ( 1 - 8 T + 66 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 33894 T^{4} - 260 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 92 T^{2} - 17562 T^{4} - 92 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 700 T^{2} + 261894 T^{4} - 700 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 196 T^{2} + 159654 T^{4} - 196 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{4} \) |
| 29 | $D_{4}$ | \( ( 1 - 20 T + 1734 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 1412 T^{2} + 2268678 T^{4} - 1412 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 3292 T^{2} + 3990 p^{2} T^{4} - 3292 p^{4} T^{6} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 44 T + 3654 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 80 T + 4098 T^{2} + 80 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 - 112 T + 7362 T^{2} - 112 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 11068 T^{2} + 46399206 T^{4} - 11068 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 462054 T^{4} - 260 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{4} \) |
| 67 | $D_{4}$ | \( ( 1 + 64 T + 7650 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 3140 T^{2} + 9051462 T^{4} - 3140 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 13372 T^{2} + 90439878 T^{4} - 13372 p^{4} T^{6} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 12548 T^{2} + 107282310 T^{4} - 12548 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 112 T + 16866 T^{2} + 112 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 156 T + 21158 T^{2} - 156 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 16124 T^{2} + 135775494 T^{4} - 16124 p^{4} T^{6} + p^{8} T^{8} \) |
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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
−6.12451941926999495325739475960, −6.03208844416074999839953101512, −5.82852508366691041337577411083, −5.48498136375244718295027976706, −5.24840496466630231820128582590, −4.99025064222820037379099235186, −4.82537805946898397638851746399, −4.77212974632791444277020069244, −4.59390084572989829555375052813, −4.46380409453654099556027639396, −4.00967924110407174899875285826, −3.85578627108723818195243741431, −3.62315504833312831281628753005, −3.40208833957559305566180680404, −3.19306688227608737086726324480, −2.82183303690798302984447902779, −2.50915735469295351129733853014, −2.25445218293342034125111943773, −1.91518501642645343230119364302, −1.87529242680191164721104249201, −1.52680392142717275354575687927, −1.30207589979741729554202849243, −1.08084890569095120765143974585, −0.46887031296433922842496471490, −0.37735649007442704438870130506,
0.37735649007442704438870130506, 0.46887031296433922842496471490, 1.08084890569095120765143974585, 1.30207589979741729554202849243, 1.52680392142717275354575687927, 1.87529242680191164721104249201, 1.91518501642645343230119364302, 2.25445218293342034125111943773, 2.50915735469295351129733853014, 2.82183303690798302984447902779, 3.19306688227608737086726324480, 3.40208833957559305566180680404, 3.62315504833312831281628753005, 3.85578627108723818195243741431, 4.00967924110407174899875285826, 4.46380409453654099556027639396, 4.59390084572989829555375052813, 4.77212974632791444277020069244, 4.82537805946898397638851746399, 4.99025064222820037379099235186, 5.24840496466630231820128582590, 5.48498136375244718295027976706, 5.82852508366691041337577411083, 6.03208844416074999839953101512, 6.12451941926999495325739475960
