| L(s) = 1 | + 2·7-s − 9-s + 12·17-s + 8·23-s + 6·25-s − 16·31-s + 20·41-s − 16·47-s + 3·49-s − 2·63-s − 8·71-s − 4·73-s + 16·79-s + 81-s − 12·89-s + 20·97-s − 28·113-s + 24·119-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 12·153-s + 157-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 1/3·9-s + 2.91·17-s + 1.66·23-s + 6/5·25-s − 2.87·31-s + 3.12·41-s − 2.33·47-s + 3/7·49-s − 0.251·63-s − 0.949·71-s − 0.468·73-s + 1.80·79-s + 1/9·81-s − 1.27·89-s + 2.03·97-s − 2.63·113-s + 2.20·119-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.970·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28901376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28901376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.212324935\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.212324935\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−8.258586685116200132210106221208, −7.88781252360304926072498015823, −7.64044413079419754780784026276, −7.45913748970519490328603070336, −6.95995482126243529990088275148, −6.72467459633671533423033595024, −6.05688184767140547700334455196, −5.66777401606308578643501100336, −5.49888218457924276521964399860, −5.24508929653651177706959449190, −4.58758262558918124527141042456, −4.57705035884816454845702540714, −3.64946635463417468760107278685, −3.56672625192291651547655128791, −2.92476887566870055165747412571, −2.90517139333556618682232802217, −2.00008127287498326774818972587, −1.58971598970734549029226699269, −1.04112517379339228921752614430, −0.63884050561526544771587866917,
0.63884050561526544771587866917, 1.04112517379339228921752614430, 1.58971598970734549029226699269, 2.00008127287498326774818972587, 2.90517139333556618682232802217, 2.92476887566870055165747412571, 3.56672625192291651547655128791, 3.64946635463417468760107278685, 4.57705035884816454845702540714, 4.58758262558918124527141042456, 5.24508929653651177706959449190, 5.49888218457924276521964399860, 5.66777401606308578643501100336, 6.05688184767140547700334455196, 6.72467459633671533423033595024, 6.95995482126243529990088275148, 7.45913748970519490328603070336, 7.64044413079419754780784026276, 7.88781252360304926072498015823, 8.258586685116200132210106221208
