| L(s) = 1 | + 6·3-s − 11·5-s + 27·9-s + 5·11-s − 5·13-s − 66·15-s − 100·17-s + 67·19-s − 76·23-s − 111·25-s + 108·27-s + 275·29-s + 362·31-s + 30·33-s − 5·37-s − 30·39-s + 162·41-s − 721·43-s − 297·45-s − 216·47-s − 600·51-s + 495·53-s − 55·55-s + 402·57-s + 173·59-s + 532·61-s + 55·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.983·5-s + 9-s + 0.137·11-s − 0.106·13-s − 1.13·15-s − 1.42·17-s + 0.808·19-s − 0.689·23-s − 0.887·25-s + 0.769·27-s + 1.76·29-s + 2.09·31-s + 0.158·33-s − 0.0222·37-s − 0.123·39-s + 0.617·41-s − 2.55·43-s − 0.983·45-s − 0.670·47-s − 1.64·51-s + 1.28·53-s − 0.134·55-s + 0.934·57-s + 0.381·59-s + 1.11·61-s + 0.104·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + 11 T + 232 T^{2} + 11 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 5 T + 304 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 3194 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 100 T + 11554 T^{2} + 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 67 T + 14406 T^{2} - 67 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 76 T + 6478 T^{2} + 76 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 275 T + 61846 T^{2} - 275 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 362 T + 73043 T^{2} - 362 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 5 T + 3606 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 162 T + 128770 T^{2} - 162 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 721 T + 288540 T^{2} + 721 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 216 T + 156778 T^{2} + 216 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 495 T + 355102 T^{2} - 495 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 173 T + 282706 T^{2} - 173 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 532 T + 447518 T^{2} - 532 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 111 T + 318532 T^{2} + 111 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1600 T + 1262410 T^{2} + 1600 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1215 T + 1011556 T^{2} + 1215 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1460 T + 1333505 T^{2} + 1460 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1409 T + 1147696 T^{2} + 1409 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1974 T + 2298994 T^{2} - 1974 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 561 T + 1863448 T^{2} + 561 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.371265433724889973001824253773, −8.304531954622511075688480929681, −7.58875644631626325749891629200, −7.51812530491013254373458673871, −6.90228987848464914718375896584, −6.75414196103584944190352366841, −6.04849359702424076022020697274, −5.94900799166495831162353147796, −4.96856226153666463224181356574, −4.77055690943627350937488380148, −4.24420278218345820367824723616, −4.09313309230251420716759533840, −3.45963951248356663503328635068, −3.10395333161220451828094653506, −2.46679314217479905467348209222, −2.39107797986518227122124598247, −1.34012716247475349775739027010, −1.22597306770555361957734617123, 0, 0,
1.22597306770555361957734617123, 1.34012716247475349775739027010, 2.39107797986518227122124598247, 2.46679314217479905467348209222, 3.10395333161220451828094653506, 3.45963951248356663503328635068, 4.09313309230251420716759533840, 4.24420278218345820367824723616, 4.77055690943627350937488380148, 4.96856226153666463224181356574, 5.94900799166495831162353147796, 6.04849359702424076022020697274, 6.75414196103584944190352366841, 6.90228987848464914718375896584, 7.51812530491013254373458673871, 7.58875644631626325749891629200, 8.304531954622511075688480929681, 8.371265433724889973001824253773
