| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s + 2·5-s + 4·6-s − 4·8-s + 3·9-s − 4·10-s − 6·12-s + 4·13-s − 4·15-s + 5·16-s − 2·17-s − 6·18-s + 8·19-s + 6·20-s − 8·23-s + 8·24-s + 3·25-s − 8·26-s − 4·27-s + 12·29-s + 8·30-s + 8·31-s − 6·32-s + 4·34-s + 9·36-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s − 1.41·8-s + 9-s − 1.26·10-s − 1.73·12-s + 1.10·13-s − 1.03·15-s + 5/4·16-s − 0.485·17-s − 1.41·18-s + 1.83·19-s + 1.34·20-s − 1.66·23-s + 1.63·24-s + 3/5·25-s − 1.56·26-s − 0.769·27-s + 2.22·29-s + 1.46·30-s + 1.43·31-s − 1.06·32-s + 0.685·34-s + 3/2·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8931120124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8931120124\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 62 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 134 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 158 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 86 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−10.82163355193668307018676417226, −10.66897452450089206237561704530, −10.12318111344345606506015049476, −9.994431475052425781178676731231, −9.306235401155395774951580444631, −9.182639654088443631277057837489, −8.406882925152879161715122905040, −8.083619211897516875950329339893, −7.53614765676431709620885770646, −7.08122970200987101916578871551, −6.33359681418963461217012528287, −6.21828911469507733362298839674, −5.84868541034693098454946060810, −5.26432646813672360798928205475, −4.48073878935550438757448272599, −3.96246986274276311927666544286, −2.82239410900099257660819290117, −2.43095958383165842560489468885, −1.22340138276186388314245985919, −0.943286290071500998448808580284,
0.943286290071500998448808580284, 1.22340138276186388314245985919, 2.43095958383165842560489468885, 2.82239410900099257660819290117, 3.96246986274276311927666544286, 4.48073878935550438757448272599, 5.26432646813672360798928205475, 5.84868541034693098454946060810, 6.21828911469507733362298839674, 6.33359681418963461217012528287, 7.08122970200987101916578871551, 7.53614765676431709620885770646, 8.083619211897516875950329339893, 8.406882925152879161715122905040, 9.182639654088443631277057837489, 9.306235401155395774951580444631, 9.994431475052425781178676731231, 10.12318111344345606506015049476, 10.66897452450089206237561704530, 10.82163355193668307018676417226
