| L(s) = 1 | + 2·5-s − 8·11-s − 25-s + 12·29-s + 8·31-s + 20·41-s − 2·49-s − 16·55-s − 8·59-s − 4·61-s + 24·79-s − 20·89-s − 4·101-s − 4·109-s + 26·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 24·145-s + 149-s + 151-s + 16·155-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 2.41·11-s − 1/5·25-s + 2.22·29-s + 1.43·31-s + 3.12·41-s − 2/7·49-s − 2.15·55-s − 1.04·59-s − 0.512·61-s + 2.70·79-s − 2.11·89-s − 0.398·101-s − 0.383·109-s + 2.36·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.99·145-s + 0.0819·149-s + 0.0813·151-s + 1.28·155-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.415486883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.415486883\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.975964025100136206621664613492, −8.444704966249952465440110312309, −8.195899155635972310779879105011, −7.80236829973923599195956978982, −7.66199205490648303859296252261, −7.11790515786410820470321984898, −6.47634634599677172449226370459, −6.38817983545297678394646740654, −5.72798517486197668433087073447, −5.67699219058747295802009781864, −5.09908016536640263023701841801, −4.63509381443500709508140664893, −4.51479370468263479328035015029, −3.84865174102725762780628519047, −2.93200345946838934148684090242, −2.88183198193585032003604208993, −2.46351656050060685852924316645, −1.99396419588046229985214049995, −1.13630905254918138951108820358, −0.53434555950747201256705787350,
0.53434555950747201256705787350, 1.13630905254918138951108820358, 1.99396419588046229985214049995, 2.46351656050060685852924316645, 2.88183198193585032003604208993, 2.93200345946838934148684090242, 3.84865174102725762780628519047, 4.51479370468263479328035015029, 4.63509381443500709508140664893, 5.09908016536640263023701841801, 5.67699219058747295802009781864, 5.72798517486197668433087073447, 6.38817983545297678394646740654, 6.47634634599677172449226370459, 7.11790515786410820470321984898, 7.66199205490648303859296252261, 7.80236829973923599195956978982, 8.195899155635972310779879105011, 8.444704966249952465440110312309, 8.975964025100136206621664613492
