| L(s) = 1 | − 7-s − 2·9-s − 6·11-s − 10·23-s − 8·25-s − 14·29-s − 2·37-s − 2·43-s + 49-s − 8·53-s + 2·63-s + 24·67-s − 8·71-s + 6·77-s + 4·79-s − 5·81-s + 12·99-s − 16·107-s + 2·109-s − 20·113-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 2/3·9-s − 1.80·11-s − 2.08·23-s − 8/5·25-s − 2.59·29-s − 0.328·37-s − 0.304·43-s + 1/7·49-s − 1.09·53-s + 0.251·63-s + 2.93·67-s − 0.949·71-s + 0.683·77-s + 0.450·79-s − 5/9·81-s + 1.20·99-s − 1.54·107-s + 0.191·109-s − 1.88·113-s + 6/11·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 48 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
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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
−7.64210051925709498480488110594, −7.27309798085250498397510618722, −6.47845207367011037225536902179, −6.19759052229895980215715590766, −5.65288805331406740896297392700, −5.34642602621472053760581125595, −5.12031329682819886447108004982, −4.08112482832077853161261599683, −3.92613333588892902951122960699, −3.35825163165326598389133443256, −2.64516975426521635316254774303, −2.20088337835097177745570204669, −1.70185143109612655884642407069, 0, 0,
1.70185143109612655884642407069, 2.20088337835097177745570204669, 2.64516975426521635316254774303, 3.35825163165326598389133443256, 3.92613333588892902951122960699, 4.08112482832077853161261599683, 5.12031329682819886447108004982, 5.34642602621472053760581125595, 5.65288805331406740896297392700, 6.19759052229895980215715590766, 6.47845207367011037225536902179, 7.27309798085250498397510618722, 7.64210051925709498480488110594
