| L(s) = 1 | + 6·3-s − 6·5-s + 16·7-s + 27·9-s + 22·11-s + 14·13-s − 36·15-s − 98·17-s − 118·19-s + 96·21-s − 6·23-s − 150·25-s + 108·27-s + 250·29-s − 88·31-s + 132·33-s − 96·35-s − 160·37-s + 84·39-s − 266·41-s − 78·43-s − 162·45-s − 250·47-s − 202·49-s − 588·51-s + 254·53-s − 132·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.536·5-s + 0.863·7-s + 9-s + 0.603·11-s + 0.298·13-s − 0.619·15-s − 1.39·17-s − 1.42·19-s + 0.997·21-s − 0.0543·23-s − 6/5·25-s + 0.769·27-s + 1.60·29-s − 0.509·31-s + 0.696·33-s − 0.463·35-s − 0.710·37-s + 0.344·39-s − 1.01·41-s − 0.276·43-s − 0.536·45-s − 0.775·47-s − 0.588·49-s − 1.61·51-s + 0.658·53-s − 0.323·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4460544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4460544 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 6 T + 186 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 16 T + 458 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 14 T + 4370 T^{2} - 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 98 T + 10402 T^{2} + 98 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 118 T + 13622 T^{2} + 118 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 7918 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 250 T + 58490 T^{2} - 250 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 88 T + 56846 T^{2} + 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 160 T + 58358 T^{2} + 160 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 266 T + 123338 T^{2} + 266 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 78 T + 154622 T^{2} + 78 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 250 T + 219694 T^{2} + 250 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 254 T + 312058 T^{2} - 254 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 388 T + 342982 T^{2} + 388 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 74 T + 455258 T^{2} + 74 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 p T + 598598 T^{2} + 8 p^{4} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 22 T + 715870 T^{2} + 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 952 T + 969278 T^{2} + 952 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 524 T + 1050050 T^{2} - 524 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1388 T + 1267510 T^{2} + 1388 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 380 T + 1277846 T^{2} + 380 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 916 T + 684902 T^{2} + 916 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.578918921187662510757663872133, −8.401177246316200551093942376088, −7.75954578960296039916753556553, −7.58507251872749159527064127246, −7.01810260816951107254155231096, −6.72366628716747047563743397568, −6.16796459434775418686970097259, −6.05745741482115483259563458106, −5.10644229465978132471663107696, −4.78270184363546641761663468600, −4.31325030163663906377447377317, −4.18383240167138708725222576512, −3.49549345924467952682366349013, −3.28570000052112846236792036717, −2.37042133764654825432198563437, −2.27550858628680322567751381844, −1.44214271329740305476364005552, −1.39129870129452046420645533996, 0, 0,
1.39129870129452046420645533996, 1.44214271329740305476364005552, 2.27550858628680322567751381844, 2.37042133764654825432198563437, 3.28570000052112846236792036717, 3.49549345924467952682366349013, 4.18383240167138708725222576512, 4.31325030163663906377447377317, 4.78270184363546641761663468600, 5.10644229465978132471663107696, 6.05745741482115483259563458106, 6.16796459434775418686970097259, 6.72366628716747047563743397568, 7.01810260816951107254155231096, 7.58507251872749159527064127246, 7.75954578960296039916753556553, 8.401177246316200551093942376088, 8.578918921187662510757663872133
