| L(s) = 1 | + 4-s − 7-s − 3·11-s + 16-s + 12·23-s + 2·25-s − 28-s + 15·29-s + 10·37-s + 4·43-s − 3·44-s − 6·49-s − 12·53-s + 64-s − 5·67-s + 12·71-s + 3·77-s + 10·79-s + 12·92-s + 2·100-s − 3·107-s + 10·109-s − 112-s − 18·113-s + 15·116-s − 13·121-s + 127-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.377·7-s − 0.904·11-s + 1/4·16-s + 2.50·23-s + 2/5·25-s − 0.188·28-s + 2.78·29-s + 1.64·37-s + 0.609·43-s − 0.452·44-s − 6/7·49-s − 1.64·53-s + 1/8·64-s − 0.610·67-s + 1.42·71-s + 0.341·77-s + 1.12·79-s + 1.25·92-s + 1/5·100-s − 0.290·107-s + 0.957·109-s − 0.0944·112-s − 1.69·113-s + 1.39·116-s − 1.18·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.465565733\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.465565733\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 109 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
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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
−7.937391776862262487108348805362, −7.68218133483775864024233892449, −6.95750116187864916688740652005, −6.70560784798031940089390938490, −6.36862857855936098839709322412, −5.91640674141710362447095436091, −5.15048987579808129503776929129, −4.93397766641927861829630882997, −4.56462103439396064097144180854, −3.79780945628844250931626369813, −3.00552827782712402555254810800, −2.88276004118655470913758768232, −2.43283859455065084464030378883, −1.34368249999935027421367604738, −0.75298744398032218528892677724,
0.75298744398032218528892677724, 1.34368249999935027421367604738, 2.43283859455065084464030378883, 2.88276004118655470913758768232, 3.00552827782712402555254810800, 3.79780945628844250931626369813, 4.56462103439396064097144180854, 4.93397766641927861829630882997, 5.15048987579808129503776929129, 5.91640674141710362447095436091, 6.36862857855936098839709322412, 6.70560784798031940089390938490, 6.95750116187864916688740652005, 7.68218133483775864024233892449, 7.937391776862262487108348805362
