| L(s) = 1 | + 6·3-s + 9·9-s + 60·11-s + 144·13-s + 288·23-s + 218·25-s − 108·27-s + 360·33-s + 144·37-s + 864·39-s − 1.15e3·47-s + 398·49-s − 828·59-s − 1.00e3·61-s + 1.72e3·69-s + 1.44e3·71-s + 356·73-s + 1.30e3·75-s − 891·81-s − 876·83-s + 1.30e3·97-s + 540·99-s − 3.32e3·107-s + 2.44e3·109-s + 864·111-s + 1.29e3·117-s + 38·121-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s + 1.64·11-s + 3.07·13-s + 2.61·23-s + 1.74·25-s − 0.769·27-s + 1.89·33-s + 0.639·37-s + 3.54·39-s − 3.57·47-s + 1.16·49-s − 1.82·59-s − 2.11·61-s + 3.01·69-s + 2.40·71-s + 0.570·73-s + 2.01·75-s − 1.22·81-s − 1.15·83-s + 1.36·97-s + 0.548·99-s − 3.00·107-s + 2.15·109-s + 0.738·111-s + 1.02·117-s + 0.0285·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.178233579\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.178233579\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - 2 p T + p^{3} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 218 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 398 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 30 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 72 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 7234 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 13070 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 144 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 48746 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 10910 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 72 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 44530 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 106526 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 576 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 32762 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 414 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 504 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 21202 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 720 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 178 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 50366 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 438 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 660850 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 650 T + p^{3} T^{2} )^{2} \) |
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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
−10.13899214341900857955868272112, −9.269344872992837976155001630935, −9.168106010111511902872344041268, −8.926741821150822672903021602055, −8.666636444021353293013619027303, −7.950495980934709103522300265100, −7.922814700948561537578784069393, −6.95045361326107877536244052875, −6.55109439563130940834450658891, −6.46675890895733500390434668570, −5.87483246301908850082338604726, −5.09953951291311930579745150150, −4.68146771312191446444942494439, −3.91034923086502229398601516015, −3.63609502191510950065329334440, −3.06772048897158497848579256964, −2.88414852632983942198080465081, −1.53821500099580977705815329629, −1.41622080603324159481921928945, −0.811579040061299962554660406386,
0.811579040061299962554660406386, 1.41622080603324159481921928945, 1.53821500099580977705815329629, 2.88414852632983942198080465081, 3.06772048897158497848579256964, 3.63609502191510950065329334440, 3.91034923086502229398601516015, 4.68146771312191446444942494439, 5.09953951291311930579745150150, 5.87483246301908850082338604726, 6.46675890895733500390434668570, 6.55109439563130940834450658891, 6.95045361326107877536244052875, 7.922814700948561537578784069393, 7.950495980934709103522300265100, 8.666636444021353293013619027303, 8.926741821150822672903021602055, 9.168106010111511902872344041268, 9.269344872992837976155001630935, 10.13899214341900857955868272112
