| L(s) = 1 | − 4-s + 5·9-s + 16-s + 2·19-s + 10·29-s + 20·31-s − 5·36-s + 4·41-s + 13·49-s + 14·59-s − 8·61-s − 64-s − 2·76-s + 20·79-s + 16·81-s + 20·89-s − 8·101-s − 26·109-s − 10·116-s − 22·121-s − 20·124-s + 127-s + 131-s + 137-s + 139-s + 5·144-s + 149-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 5/3·9-s + 1/4·16-s + 0.458·19-s + 1.85·29-s + 3.59·31-s − 5/6·36-s + 0.624·41-s + 13/7·49-s + 1.82·59-s − 1.02·61-s − 1/8·64-s − 0.229·76-s + 2.25·79-s + 16/9·81-s + 2.11·89-s − 0.796·101-s − 2.49·109-s − 0.928·116-s − 2·121-s − 1.79·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5/12·144-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.607958809\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.607958809\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−10.18045213609076050203781860461, −9.887229082940113276627946832169, −9.521732143685579722395560465754, −9.040084914755131211440845574866, −8.570767921482245233489888072917, −8.146375880621783584188629201985, −7.73536537914502029469829826164, −7.42322210336377208774835263203, −6.69605789025234536577527003526, −6.48597861289769798939981143583, −6.16558918619354528010285186770, −5.23414264969154629438542205541, −4.93342210299783507043727648191, −4.53755516569122730922776405965, −4.00081605933044469909938997732, −3.68868125335052577976451741625, −2.58516429259994352815771479682, −2.54915768872178716372889155151, −1.07694327410406104773496674464, −1.06935986225685257853468118180,
1.06935986225685257853468118180, 1.07694327410406104773496674464, 2.54915768872178716372889155151, 2.58516429259994352815771479682, 3.68868125335052577976451741625, 4.00081605933044469909938997732, 4.53755516569122730922776405965, 4.93342210299783507043727648191, 5.23414264969154629438542205541, 6.16558918619354528010285186770, 6.48597861289769798939981143583, 6.69605789025234536577527003526, 7.42322210336377208774835263203, 7.73536537914502029469829826164, 8.146375880621783584188629201985, 8.570767921482245233489888072917, 9.040084914755131211440845574866, 9.521732143685579722395560465754, 9.887229082940113276627946832169, 10.18045213609076050203781860461
