| L(s) = 1 | − 2·4-s + 2·5-s − 4·7-s − 9-s − 4·11-s + 4·16-s − 4·20-s − 25-s + 8·28-s − 8·31-s − 8·35-s + 2·36-s + 8·43-s + 8·44-s − 2·45-s − 8·47-s + 9·49-s − 8·55-s + 4·61-s + 4·63-s − 8·64-s − 8·67-s + 16·77-s + 8·80-s + 81-s + 4·99-s + 2·100-s + ⋯ |
| L(s) = 1 | − 4-s + 0.894·5-s − 1.51·7-s − 1/3·9-s − 1.20·11-s + 16-s − 0.894·20-s − 1/5·25-s + 1.51·28-s − 1.43·31-s − 1.35·35-s + 1/3·36-s + 1.21·43-s + 1.20·44-s − 0.298·45-s − 1.16·47-s + 9/7·49-s − 1.07·55-s + 0.512·61-s + 0.503·63-s − 64-s − 0.977·67-s + 1.82·77-s + 0.894·80-s + 1/9·81-s + 0.402·99-s + 1/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5733084982\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5733084982\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
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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
−8.387468451873275365546050862032, −7.83748705443579405196308839687, −7.55442873541508846215560967501, −6.90581891622102727236158300363, −6.42881276266002124399775592991, −5.93223626244332950775664967658, −5.56123612153685381908783666990, −5.28902875655497949426091209455, −4.66205478776421038273850178059, −3.96903393507620593117594217641, −3.53702168990738753868047559890, −2.91102797945808126924573884820, −2.48474336764674253641146661654, −1.59287391495841526090039867954, −0.36857824903111444097348721185,
0.36857824903111444097348721185, 1.59287391495841526090039867954, 2.48474336764674253641146661654, 2.91102797945808126924573884820, 3.53702168990738753868047559890, 3.96903393507620593117594217641, 4.66205478776421038273850178059, 5.28902875655497949426091209455, 5.56123612153685381908783666990, 5.93223626244332950775664967658, 6.42881276266002124399775592991, 6.90581891622102727236158300363, 7.55442873541508846215560967501, 7.83748705443579405196308839687, 8.387468451873275365546050862032
