| 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
−8.418825520064818194527514901346, −8.255370637075054174168104316417, −7.57541768564203921234570226091, −7.42523035876584311442121348156, −6.75953173066841733738456494869, −6.69330370584871247481369240060, −6.23031523892482162500099139452, −5.79385850286664841185871134720, −5.33189137179598006632870756844, −5.06145281662906126109485564353, −4.42282214756549885799793828803, −4.26466877378715155919929278896, −3.43019755457225140365630025388, −3.39036377294537224213491653308, −2.53916943186236624610968690897, −2.17443433416504173461024927127, −1.27819071377316072889279350106, −0.847399052067183258034641029619, 0, 0,
0.847399052067183258034641029619, 1.27819071377316072889279350106, 2.17443433416504173461024927127, 2.53916943186236624610968690897, 3.39036377294537224213491653308, 3.43019755457225140365630025388, 4.26466877378715155919929278896, 4.42282214756549885799793828803, 5.06145281662906126109485564353, 5.33189137179598006632870756844, 5.79385850286664841185871134720, 6.23031523892482162500099139452, 6.69330370584871247481369240060, 6.75953173066841733738456494869, 7.42523035876584311442121348156, 7.57541768564203921234570226091, 8.255370637075054174168104316417, 8.418825520064818194527514901346
