| L(s) = 1 | + 512·2-s + 1.76e4·3-s + 1.96e5·4-s − 7.81e5·5-s + 9.02e6·6-s + 2.76e7·7-s + 6.71e7·8-s + 1.44e8·9-s − 4.00e8·10-s + 6.45e7·11-s + 3.46e9·12-s + 2.89e9·13-s + 1.41e10·14-s − 1.37e10·15-s + 2.14e10·16-s + 1.58e9·17-s + 7.38e10·18-s + 4.42e10·19-s − 1.53e11·20-s + 4.88e11·21-s + 3.30e10·22-s + 4.87e11·23-s + 1.18e12·24-s + 4.57e11·25-s + 1.48e12·26-s + 1.88e12·27-s + 5.44e12·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.55·3-s + 3/2·4-s − 0.894·5-s + 2.19·6-s + 1.81·7-s + 1.41·8-s + 1.11·9-s − 1.26·10-s + 0.0907·11-s + 2.32·12-s + 0.984·13-s + 2.56·14-s − 1.38·15-s + 5/4·16-s + 0.0549·17-s + 1.57·18-s + 0.597·19-s − 1.34·20-s + 2.81·21-s + 0.128·22-s + 1.29·23-s + 2.19·24-s + 3/5·25-s + 1.39·26-s + 1.28·27-s + 2.72·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+17/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(14.70281786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.70281786\) |
| \(L(\frac{19}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{8} T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + p^{8} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 5876 p T + 685454 p^{5} T^{2} - 5876 p^{18} T^{3} + p^{34} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 27684196 T + 93506538073374 p T^{2} - 27684196 p^{17} T^{3} + p^{34} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 533184 p^{2} T - 3200366345168954 p^{2} T^{2} - 533184 p^{19} T^{3} + p^{34} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2895838468 T + 1418824834264307094 p T^{2} - 2895838468 p^{17} T^{3} + p^{34} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 1580212596 T + \)\(13\!\cdots\!58\)\( T^{2} - 1580212596 p^{17} T^{3} + p^{34} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 44213712760 T + \)\(27\!\cdots\!78\)\( T^{2} - 44213712760 p^{17} T^{3} + p^{34} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 487549782828 T + \)\(24\!\cdots\!02\)\( T^{2} - 487549782828 p^{17} T^{3} + p^{34} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3987314863500 T + \)\(18\!\cdots\!18\)\( T^{2} - 3987314863500 p^{17} T^{3} + p^{34} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 5492261339336 T + \)\(52\!\cdots\!46\)\( T^{2} + 5492261339336 p^{17} T^{3} + p^{34} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 62715287637884 T + \)\(18\!\cdots\!98\)\( T^{2} + 62715287637884 p^{17} T^{3} + p^{34} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 23411477277324 T + \)\(10\!\cdots\!06\)\( T^{2} - 23411477277324 p^{17} T^{3} + p^{34} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 124856923191092 T + \)\(14\!\cdots\!02\)\( T^{2} + 124856923191092 p^{17} T^{3} + p^{34} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 185946612123564 T + \)\(57\!\cdots\!98\)\( T^{2} + 185946612123564 p^{17} T^{3} + p^{34} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 359339780647668 T + \)\(33\!\cdots\!82\)\( T^{2} - 359339780647668 p^{17} T^{3} + p^{34} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 902179170360600 T + \)\(20\!\cdots\!38\)\( T^{2} - 902179170360600 p^{17} T^{3} + p^{34} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1564422918967676 T + \)\(36\!\cdots\!86\)\( T^{2} + 1564422918967676 p^{17} T^{3} + p^{34} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 5839738931054684 T + \)\(30\!\cdots\!18\)\( T^{2} + 5839738931054684 p^{17} T^{3} + p^{34} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 67588560434136 T + \)\(50\!\cdots\!06\)\( T^{2} + 67588560434136 p^{17} T^{3} + p^{34} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 3533390699585668 T + \)\(90\!\cdots\!62\)\( T^{2} - 3533390699585668 p^{17} T^{3} + p^{34} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 19002656396552080 T + \)\(43\!\cdots\!18\)\( T^{2} - 19002656396552080 p^{17} T^{3} + p^{34} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 261145638254436 p T + \)\(75\!\cdots\!82\)\( T^{2} - 261145638254436 p^{18} T^{3} + p^{34} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 97499522192222220 T + \)\(48\!\cdots\!58\)\( T^{2} + 97499522192222220 p^{17} T^{3} + p^{34} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 99889937855386756 T + \)\(14\!\cdots\!58\)\( T^{2} - 99889937855386756 p^{17} T^{3} + p^{34} T^{4} \) |
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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
−16.49452189855450298359084947718, −15.79977185819673406160331055604, −14.91810322526225841113093940138, −14.89257222253528925539165822730, −13.87900508727161083944734268257, −13.82863554692487995426831217217, −12.66806956792330355632198207896, −11.85444941196240143799885967530, −11.21093525111772113575974931888, −10.48194131361407213808657194346, −8.716239185577053457074559797551, −8.414702524300610678224362229570, −7.56453926402556833448452837594, −6.76098960845187442525068610138, −5.11636671786388053697245566920, −4.66794664716345343582458429224, −3.48709245950675540928065591627, −3.17904426674912031285594886645, −1.90031851112469128941490957302, −1.22162989460489883800798337344,
1.22162989460489883800798337344, 1.90031851112469128941490957302, 3.17904426674912031285594886645, 3.48709245950675540928065591627, 4.66794664716345343582458429224, 5.11636671786388053697245566920, 6.76098960845187442525068610138, 7.56453926402556833448452837594, 8.414702524300610678224362229570, 8.716239185577053457074559797551, 10.48194131361407213808657194346, 11.21093525111772113575974931888, 11.85444941196240143799885967530, 12.66806956792330355632198207896, 13.82863554692487995426831217217, 13.87900508727161083944734268257, 14.89257222253528925539165822730, 14.91810322526225841113093940138, 15.79977185819673406160331055604, 16.49452189855450298359084947718
