| L(s) = 1 | + 6·3-s + 6·5-s − 22·7-s + 27·9-s − 22·11-s − 94·13-s + 36·15-s + 56·17-s + 76·19-s − 132·21-s − 54·23-s − 38·25-s + 108·27-s − 104·29-s − 224·31-s − 132·33-s − 132·35-s + 68·37-s − 564·39-s − 300·41-s + 456·43-s + 162·45-s + 22·47-s − 138·49-s + 336·51-s + 110·53-s − 132·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.536·5-s − 1.18·7-s + 9-s − 0.603·11-s − 2.00·13-s + 0.619·15-s + 0.798·17-s + 0.917·19-s − 1.37·21-s − 0.489·23-s − 0.303·25-s + 0.769·27-s − 0.665·29-s − 1.29·31-s − 0.696·33-s − 0.637·35-s + 0.302·37-s − 2.31·39-s − 1.14·41-s + 1.61·43-s + 0.536·45-s + 0.0682·47-s − 0.402·49-s + 0.922·51-s + 0.285·53-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
| \(L(2)\) |
\(\approx\) |
\(2.321391568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.321391568\) |
| \(L(\frac{5}{2})\) |
|
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 | $D_{4}$ | \( 1 - 6 T + 74 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 22 T + 622 T^{2} + 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 94 T + 6418 T^{2} + 94 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 56 T + 3950 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 p T + 8502 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 54 T + 20438 T^{2} + 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 104 T + 50742 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 224 T + 60286 T^{2} + 224 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 68 T + 55102 T^{2} - 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 300 T + 157382 T^{2} + 300 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 456 T + 208038 T^{2} - 456 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 22 T + 72902 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 110 T + 74154 T^{2} - 110 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 68 T + 322374 T^{2} - 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 54 T + 454506 T^{2} - 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 1560 T + 1198086 T^{2} - 1560 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 274 T + 692966 T^{2} - 274 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 548 T + 663670 T^{2} - 548 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 922 T + 1100734 T^{2} + 922 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1184 T + 1135878 T^{2} - 1184 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 956 T + 679382 T^{2} + 956 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 672 T + 1031742 T^{2} + 672 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.006221783564908536812975500931, −8.643228878533511575374818740996, −8.046225130277898102345592806619, −7.70839068304532979820595563828, −7.48461483457504842699285591986, −7.16183229612325246703056891749, −6.64820631922748172745847571862, −6.33028731036095149710527014873, −5.60802437522564741012852849957, −5.34828764732895783530844109684, −5.10266799774722743064144885571, −4.42827423868580792285873190322, −3.77721440989544171153465386120, −3.59646864208420516932816991379, −2.95759497037156844577357184458, −2.72353934390898601065988138540, −2.05116833688281734041405578419, −1.95019949478616471911908507444, −0.940157920426247173825464877326, −0.30615693752282827744041172649,
0.30615693752282827744041172649, 0.940157920426247173825464877326, 1.95019949478616471911908507444, 2.05116833688281734041405578419, 2.72353934390898601065988138540, 2.95759497037156844577357184458, 3.59646864208420516932816991379, 3.77721440989544171153465386120, 4.42827423868580792285873190322, 5.10266799774722743064144885571, 5.34828764732895783530844109684, 5.60802437522564741012852849957, 6.33028731036095149710527014873, 6.64820631922748172745847571862, 7.16183229612325246703056891749, 7.48461483457504842699285591986, 7.70839068304532979820595563828, 8.046225130277898102345592806619, 8.643228878533511575374818740996, 9.006221783564908536812975500931
