| L(s) = 1 | − 3·2-s − 3·3-s + 4·4-s − 6·5-s + 9·6-s − 7-s − 3·8-s + 2·9-s + 18·10-s − 12·12-s − 7·13-s + 3·14-s + 18·15-s + 3·16-s − 6·17-s − 6·18-s + 19-s − 24·20-s + 3·21-s + 6·23-s + 9·24-s + 17·25-s + 21·26-s + 6·27-s − 4·28-s − 6·29-s − 54·30-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 2·4-s − 2.68·5-s + 3.67·6-s − 0.377·7-s − 1.06·8-s + 2/3·9-s + 5.69·10-s − 3.46·12-s − 1.94·13-s + 0.801·14-s + 4.64·15-s + 3/4·16-s − 1.45·17-s − 1.41·18-s + 0.229·19-s − 5.36·20-s + 0.654·21-s + 1.25·23-s + 1.83·24-s + 17/5·25-s + 4.11·26-s + 1.15·27-s − 0.755·28-s − 1.11·29-s − 9.85·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4489 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 67 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 7 T + 27 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - T + 27 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + T + 63 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 83 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3 T + 77 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 15 T + 149 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 7 T + 33 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 173 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 7 T + 69 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 15 T + 161 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 173 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−14.91523776608311472957591545000, −14.42461883890019643490162312229, −12.81620168150096520907707440100, −12.67161903629462888514149979353, −11.83574552672041175960176229935, −11.56001067041141741538524132919, −11.14780217576772067169655962237, −10.85651290875254975429531714035, −9.927438659386322345236336345545, −9.419204322289150007108477222310, −8.640881423482157646777216209235, −8.203452589679141607491053854413, −7.35273556158538514937136442147, −7.26795772043444337656409462434, −6.36470697660250855000839936046, −5.11384122876646128717095438968, −4.54764937213672411163862912737, −3.25253030362544283348529857708, 0, 0,
3.25253030362544283348529857708, 4.54764937213672411163862912737, 5.11384122876646128717095438968, 6.36470697660250855000839936046, 7.26795772043444337656409462434, 7.35273556158538514937136442147, 8.203452589679141607491053854413, 8.640881423482157646777216209235, 9.419204322289150007108477222310, 9.927438659386322345236336345545, 10.85651290875254975429531714035, 11.14780217576772067169655962237, 11.56001067041141741538524132919, 11.83574552672041175960176229935, 12.67161903629462888514149979353, 12.81620168150096520907707440100, 14.42461883890019643490162312229, 14.91523776608311472957591545000
