| L(s) = 1 | + 4·7-s + 6·9-s − 12·17-s + 12·23-s − 25-s − 8·31-s + 4·41-s + 20·47-s − 2·49-s + 24·63-s + 32·71-s + 12·73-s + 27·81-s − 12·89-s + 4·97-s − 36·103-s − 12·113-s − 48·119-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 72·153-s + 157-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 2·9-s − 2.91·17-s + 2.50·23-s − 1/5·25-s − 1.43·31-s + 0.624·41-s + 2.91·47-s − 2/7·49-s + 3.02·63-s + 3.79·71-s + 1.40·73-s + 3·81-s − 1.27·89-s + 0.406·97-s − 3.54·103-s − 1.12·113-s − 4.40·119-s − 1.27·121-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 − 5.82·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.303579605\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.303579605\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.522766925008265581708437593125, −9.493747392754943732798619363952, −9.186335777498533122344301869518, −8.703725342634972309405121231116, −8.119364025750308367764127391351, −7.997887186364500056843614143035, −7.12103692032034062023407522426, −7.08411819793650163112481330124, −6.85870078974112762240265129353, −6.32700247102404542113016477337, −5.34084807863291790297727628433, −5.30132599216499067639639103420, −4.58378362597523053160330035971, −4.41056792686585751428011666231, −4.04873752738642679122680309474, −3.40009431958255309157956313088, −2.29943710976370703740471020037, −2.21732023823499989555838834279, −1.47020411563746099608139675182, −0.830256387377360005993918644132,
0.830256387377360005993918644132, 1.47020411563746099608139675182, 2.21732023823499989555838834279, 2.29943710976370703740471020037, 3.40009431958255309157956313088, 4.04873752738642679122680309474, 4.41056792686585751428011666231, 4.58378362597523053160330035971, 5.30132599216499067639639103420, 5.34084807863291790297727628433, 6.32700247102404542113016477337, 6.85870078974112762240265129353, 7.08411819793650163112481330124, 7.12103692032034062023407522426, 7.997887186364500056843614143035, 8.119364025750308367764127391351, 8.703725342634972309405121231116, 9.186335777498533122344301869518, 9.493747392754943732798619363952, 9.522766925008265581708437593125
