| L(s) = 1 | − 4-s − 2·5-s − 9-s − 8·11-s + 16-s + 2·19-s + 2·20-s − 25-s + 4·29-s − 12·31-s + 36-s + 8·44-s + 2·45-s + 14·49-s + 16·55-s − 4·59-s − 12·61-s − 64-s − 24·71-s − 2·76-s + 28·79-s − 2·80-s + 81-s − 8·89-s − 4·95-s + 8·99-s + 100-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.894·5-s − 1/3·9-s − 2.41·11-s + 1/4·16-s + 0.458·19-s + 0.447·20-s − 1/5·25-s + 0.742·29-s − 2.15·31-s + 1/6·36-s + 1.20·44-s + 0.298·45-s + 2·49-s + 2.15·55-s − 0.520·59-s − 1.53·61-s − 1/8·64-s − 2.84·71-s − 0.229·76-s + 3.15·79-s − 0.223·80-s + 1/9·81-s − 0.847·89-s − 0.410·95-s + 0.804·99-s + 1/10·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4506581117\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4506581117\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−10.76378627726636236112099785040, −10.46462821165226532758204302613, −10.42780428767633370412130043710, −9.517752060107356672020385457221, −9.210128340046148514239332393854, −8.766363184929676765386913323261, −8.106015319848450357798688585191, −7.930333423499553177034582419579, −7.49629294753819375148836117429, −7.20021440091719279360429174353, −6.40534708988336544380642055374, −5.68731223812146267683738564969, −5.34485935136836988743198872688, −5.07455768381776820349955560035, −4.22696197960079737462113218221, −3.93427986242902300787553498051, −2.96184052688672467759281624440, −2.83488330187070679192426662326, −1.79229596142358938527930914928, −0.37345075326660318719524411743,
0.37345075326660318719524411743, 1.79229596142358938527930914928, 2.83488330187070679192426662326, 2.96184052688672467759281624440, 3.93427986242902300787553498051, 4.22696197960079737462113218221, 5.07455768381776820349955560035, 5.34485935136836988743198872688, 5.68731223812146267683738564969, 6.40534708988336544380642055374, 7.20021440091719279360429174353, 7.49629294753819375148836117429, 7.930333423499553177034582419579, 8.106015319848450357798688585191, 8.766363184929676765386913323261, 9.210128340046148514239332393854, 9.517752060107356672020385457221, 10.42780428767633370412130043710, 10.46462821165226532758204302613, 10.76378627726636236112099785040
