| L(s) = 1 | − 13·2-s − 486·3-s − 3.51e3·4-s − 6.25e3·5-s + 6.31e3·6-s + 7.78e3·7-s + 6.70e4·8-s + 1.77e5·9-s + 8.12e4·10-s + 2.95e5·11-s + 1.71e6·12-s + 6.57e5·13-s − 1.01e5·14-s + 3.03e6·15-s + 8.25e6·16-s + 8.57e6·17-s − 2.30e6·18-s + 1.76e7·19-s + 2.19e7·20-s − 3.78e6·21-s − 3.84e6·22-s − 2.98e7·23-s − 3.25e7·24-s + 2.92e7·25-s − 8.54e6·26-s − 5.73e7·27-s − 2.73e7·28-s + ⋯ |
| L(s) = 1 | − 0.287·2-s − 1.15·3-s − 1.71·4-s − 0.894·5-s + 0.331·6-s + 0.175·7-s + 0.723·8-s + 9-s + 0.256·10-s + 0.553·11-s + 1.98·12-s + 0.491·13-s − 0.0502·14-s + 1.03·15-s + 1.96·16-s + 1.46·17-s − 0.287·18-s + 1.63·19-s + 1.53·20-s − 0.202·21-s − 0.158·22-s − 0.966·23-s − 0.835·24-s + 3/5·25-s − 0.141·26-s − 0.769·27-s − 0.300·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.7644370390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7644370390\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + p^{5} T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + p^{5} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + 13 T + 461 p^{3} T^{2} + 13 p^{11} T^{3} + p^{22} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 1112 p T - 37013986 p^{2} T^{2} - 1112 p^{12} T^{3} + p^{22} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 295568 T + 160442914534 T^{2} - 295568 p^{11} T^{3} + p^{22} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 657492 T + 1362446380366 T^{2} - 657492 p^{11} T^{3} + p^{22} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 8579948 T + 84202726534342 T^{2} - 8579948 p^{11} T^{3} + p^{22} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 17627976 T + 302871116598118 T^{2} - 17627976 p^{11} T^{3} + p^{22} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 29841072 T + 1016542384821454 T^{2} + 29841072 p^{11} T^{3} + p^{22} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 201881948 T + 26564448133170958 T^{2} + 201881948 p^{11} T^{3} + p^{22} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 71057008 T + 33404626522449662 T^{2} + 71057008 p^{11} T^{3} + p^{22} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 705858484 T + 10980212422450758 p T^{2} - 705858484 p^{11} T^{3} + p^{22} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 327655148 T + 975285625454018902 T^{2} + 327655148 p^{11} T^{3} + p^{22} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3192552120 T + 4401027789107914870 T^{2} - 3192552120 p^{11} T^{3} + p^{22} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2053064720 T + 5969854595602269790 T^{2} - 2053064720 p^{11} T^{3} + p^{22} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2304299452 T + 14223212019195836254 T^{2} + 2304299452 p^{11} T^{3} + p^{22} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 1478770576 T + 34879178115908872198 T^{2} + 1478770576 p^{11} T^{3} + p^{22} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8264891460 T + 97961878141180941838 T^{2} + 8264891460 p^{11} T^{3} + p^{22} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 361375784 p T + \)\(37\!\cdots\!62\)\( T^{2} - 361375784 p^{12} T^{3} + p^{22} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 20218888256 T + \)\(25\!\cdots\!26\)\( T^{2} + 20218888256 p^{11} T^{3} + p^{22} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 25879583268 T + \)\(42\!\cdots\!94\)\( T^{2} - 25879583268 p^{11} T^{3} + p^{22} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 22324995440 T + \)\(14\!\cdots\!58\)\( T^{2} - 22324995440 p^{11} T^{3} + p^{22} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 48014508984 T + \)\(30\!\cdots\!22\)\( T^{2} - 48014508984 p^{11} T^{3} + p^{22} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 79209683076 T + \)\(58\!\cdots\!98\)\( T^{2} - 79209683076 p^{11} T^{3} + p^{22} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 37075227452 T + \)\(14\!\cdots\!82\)\( T^{2} + 37075227452 p^{11} T^{3} + p^{22} T^{4} \) |
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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
−17.00060518486258215179990234521, −16.50717456802371117733134398414, −15.89747561368340138259411776309, −14.88770711314333628657448618600, −14.14379926331944508923509481931, −13.60801005543520109249561456787, −12.52040080732156116522683679583, −12.23966612941843115158081587001, −11.31649929805415295241948919255, −10.61436274528338911663302044329, −9.476870678758022322826466147871, −9.282956412132620153395400101089, −7.77524513218051268282626188862, −7.64125751743185303938353441161, −5.85383030287690847270410174135, −5.31564481096576657272496312398, −4.16645937042802381379698966810, −3.70619148320679225681175072859, −1.07452014889870494921406952175, −0.60381703179539193221583408932,
0.60381703179539193221583408932, 1.07452014889870494921406952175, 3.70619148320679225681175072859, 4.16645937042802381379698966810, 5.31564481096576657272496312398, 5.85383030287690847270410174135, 7.64125751743185303938353441161, 7.77524513218051268282626188862, 9.282956412132620153395400101089, 9.476870678758022322826466147871, 10.61436274528338911663302044329, 11.31649929805415295241948919255, 12.23966612941843115158081587001, 12.52040080732156116522683679583, 13.60801005543520109249561456787, 14.14379926331944508923509481931, 14.88770711314333628657448618600, 15.89747561368340138259411776309, 16.50717456802371117733134398414, 17.00060518486258215179990234521
