| L(s) = 1 | + 8·7-s + 5·9-s + 2·17-s + 8·23-s + 8·31-s + 6·41-s + 34·49-s + 40·63-s − 32·71-s − 22·73-s + 8·79-s + 16·81-s + 26·89-s + 28·97-s − 32·103-s − 18·113-s + 16·119-s + 13·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 10·153-s + 157-s + 64·161-s + ⋯ |
| L(s) = 1 | + 3.02·7-s + 5/3·9-s + 0.485·17-s + 1.66·23-s + 1.43·31-s + 0.937·41-s + 34/7·49-s + 5.03·63-s − 3.79·71-s − 2.57·73-s + 0.900·79-s + 16/9·81-s + 2.75·89-s + 2.84·97-s − 3.15·103-s − 1.69·113-s + 1.46·119-s + 1.18·121-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 + 0.808·153-s + 0.0798·157-s + 5.04·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.643464634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.643464634\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 165 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.914815206284655907453304454005, −8.282354013491871430601513402853, −8.006399456635320459510551428068, −7.87018976770035618399839418847, −7.32608330786106190180451574935, −7.22141428311645768783918999340, −6.77759487964778405566023454124, −6.19036224680322808017807985565, −5.69515414772576721078801219811, −5.33089022352832281038702968692, −4.80527064542830406375237522027, −4.62434033034052160430100928675, −4.41192172079942169613829064925, −4.02902320048584639973242993028, −3.23599255119938848603919489012, −2.78346627869216571351558923491, −2.08089019255575196564556233513, −1.68598405271492766436200814779, −1.15715439296928360738055919022, −1.00472086545910297191693102776,
1.00472086545910297191693102776, 1.15715439296928360738055919022, 1.68598405271492766436200814779, 2.08089019255575196564556233513, 2.78346627869216571351558923491, 3.23599255119938848603919489012, 4.02902320048584639973242993028, 4.41192172079942169613829064925, 4.62434033034052160430100928675, 4.80527064542830406375237522027, 5.33089022352832281038702968692, 5.69515414772576721078801219811, 6.19036224680322808017807985565, 6.77759487964778405566023454124, 7.22141428311645768783918999340, 7.32608330786106190180451574935, 7.87018976770035618399839418847, 8.006399456635320459510551428068, 8.282354013491871430601513402853, 8.914815206284655907453304454005
