| L(s) = 1 | − 5-s + 6·9-s + 25-s − 4·29-s + 12·41-s − 6·45-s − 2·49-s + 4·61-s + 27·81-s − 12·89-s + 12·101-s − 28·109-s − 6·121-s − 125-s + 127-s + 131-s + 137-s + 139-s + 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 2·9-s + 1/5·25-s − 0.742·29-s + 1.87·41-s − 0.894·45-s − 2/7·49-s + 0.512·61-s + 3·81-s − 1.27·89-s + 1.19·101-s − 2.68·109-s − 0.545·121-s − 0.0894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.332·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.69·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.340249496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.340249496\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−10.45635555895123398508616992264, −9.932807738951225402896384329207, −9.440574375670347063720129211679, −9.067838966976968249797003880438, −8.176970332316499283761017873268, −7.76702746734900361151457450826, −7.20137597437398225607662382241, −6.85444202581638370159310745997, −6.12074514811380938963071814274, −5.36548914818550921723419960819, −4.57309836509969959885811026572, −4.14489874143322088321845955968, −3.51023978075085753955621242562, −2.34824788381240391505005522266, −1.26603240808509178112653247284,
1.26603240808509178112653247284, 2.34824788381240391505005522266, 3.51023978075085753955621242562, 4.14489874143322088321845955968, 4.57309836509969959885811026572, 5.36548914818550921723419960819, 6.12074514811380938963071814274, 6.85444202581638370159310745997, 7.20137597437398225607662382241, 7.76702746734900361151457450826, 8.176970332316499283761017873268, 9.067838966976968249797003880438, 9.440574375670347063720129211679, 9.932807738951225402896384329207, 10.45635555895123398508616992264
