| L(s) = 1 | + 64·2-s + 604·3-s + 3.07e3·4-s + 6.25e3·5-s + 3.86e4·6-s + 1.40e4·7-s + 1.31e5·8-s + 1.16e5·9-s + 4.00e5·10-s + 4.21e5·11-s + 1.85e6·12-s + 1.73e6·13-s + 9.01e5·14-s + 3.77e6·15-s + 5.24e6·16-s − 6.32e6·17-s + 7.43e6·18-s − 2.88e7·19-s + 1.92e7·20-s + 8.51e6·21-s + 2.69e7·22-s − 4.52e7·23-s + 7.91e7·24-s + 2.92e7·25-s + 1.10e8·26-s + 2.70e7·27-s + 4.32e7·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.43·3-s + 3/2·4-s + 0.894·5-s + 2.02·6-s + 0.316·7-s + 1.41·8-s + 0.656·9-s + 1.26·10-s + 0.789·11-s + 2.15·12-s + 1.29·13-s + 0.448·14-s + 1.28·15-s + 5/4·16-s − 1.08·17-s + 0.927·18-s − 2.67·19-s + 1.34·20-s + 0.454·21-s + 1.11·22-s − 1.46·23-s + 2.02·24-s + 3/5·25-s + 1.82·26-s + 0.362·27-s + 0.475·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(10.36762341\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.36762341\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - p^{5} T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - p^{5} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 604 T + 27622 p^{2} T^{2} - 604 p^{11} T^{3} + p^{22} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 14092 T - 309197214 p T^{2} - 14092 p^{11} T^{3} + p^{22} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 421584 T + 305428208086 T^{2} - 421584 p^{11} T^{3} + p^{22} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 1730524 T + 3917874059118 T^{2} - 1730524 p^{11} T^{3} + p^{22} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6323628 T + 73563074016262 T^{2} + 6323628 p^{11} T^{3} + p^{22} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 28897400 T + 430951545856038 T^{2} + 28897400 p^{11} T^{3} + p^{22} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 45236076 T + 2116464952736398 T^{2} + 45236076 p^{11} T^{3} + p^{22} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 58226220 T + 22969270731722158 T^{2} - 58226220 p^{11} T^{3} + p^{22} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 41413384 T + 48711797584420926 T^{2} - 41413384 p^{11} T^{3} + p^{22} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 377255452 T + 373315553507530302 T^{2} - 377255452 p^{11} T^{3} + p^{22} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 785271036 T + 985831391781991606 T^{2} + 785271036 p^{11} T^{3} + p^{22} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 1452987236 T + 2194695988642931238 T^{2} + 1452987236 p^{11} T^{3} + p^{22} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 1288127748 T + 4501637759066496382 T^{2} + 1288127748 p^{11} T^{3} + p^{22} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 30490836 T + 1934178302201224318 T^{2} + 30490836 p^{11} T^{3} + p^{22} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8677102440 T + 78821502141035647318 T^{2} - 8677102440 p^{11} T^{3} + p^{22} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 1115498764 T - 16676167592870147154 T^{2} - 1115498764 p^{11} T^{3} + p^{22} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12673769708 T + \)\(18\!\cdots\!82\)\( T^{2} + 12673769708 p^{11} T^{3} + p^{22} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 13799832984 T + \)\(49\!\cdots\!06\)\( T^{2} - 13799832984 p^{11} T^{3} + p^{22} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 17842079516 T + \)\(53\!\cdots\!18\)\( T^{2} + 17842079516 p^{11} T^{3} + p^{22} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12636930320 T + \)\(13\!\cdots\!58\)\( T^{2} + 12636930320 p^{11} T^{3} + p^{22} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 41986488924 T + \)\(18\!\cdots\!78\)\( T^{2} - 41986488924 p^{11} T^{3} + p^{22} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 13208740020 T + \)\(53\!\cdots\!78\)\( T^{2} - 13208740020 p^{11} T^{3} + p^{22} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 61787462828 T + \)\(91\!\cdots\!02\)\( T^{2} + 61787462828 p^{11} T^{3} + p^{22} T^{4} \) |
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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
−18.69748957891774686927420549159, −17.77736466117883358148823341509, −16.93640620326652268169203828394, −16.08971783516549348062018525335, −15.08580677503117582957814296325, −14.75357021979853328790989824818, −14.07432227422831341256009847652, −13.43059715146488194197007047003, −13.11436772873243776789052969339, −11.97272180346616251378113402968, −10.97230457851962835483947881259, −10.11655535441403759245036547554, −8.638804591976078545783897909957, −8.436532141699225614016171337797, −6.60100364447373069342408599248, −6.15015776511802460800897197579, −4.50776396697386038857954667873, −3.70379288908772639419701663880, −2.40946196629544146143956038996, −1.77322034821727295348141717469,
1.77322034821727295348141717469, 2.40946196629544146143956038996, 3.70379288908772639419701663880, 4.50776396697386038857954667873, 6.15015776511802460800897197579, 6.60100364447373069342408599248, 8.436532141699225614016171337797, 8.638804591976078545783897909957, 10.11655535441403759245036547554, 10.97230457851962835483947881259, 11.97272180346616251378113402968, 13.11436772873243776789052969339, 13.43059715146488194197007047003, 14.07432227422831341256009847652, 14.75357021979853328790989824818, 15.08580677503117582957814296325, 16.08971783516549348062018525335, 16.93640620326652268169203828394, 17.77736466117883358148823341509, 18.69748957891774686927420549159
