| L(s) = 1 | − 2·3-s + 2·7-s + 2·9-s − 8·13-s − 8·17-s + 8·19-s − 4·21-s + 10·23-s − 6·27-s + 16·39-s − 8·41-s − 14·43-s + 6·47-s + 2·49-s + 16·51-s − 8·53-s − 16·57-s + 8·59-s + 16·61-s + 4·63-s + 6·67-s − 20·69-s + 8·73-s + 16·79-s + 11·81-s + 10·83-s − 16·91-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.755·7-s + 2/3·9-s − 2.21·13-s − 1.94·17-s + 1.83·19-s − 0.872·21-s + 2.08·23-s − 1.15·27-s + 2.56·39-s − 1.24·41-s − 2.13·43-s + 0.875·47-s + 2/7·49-s + 2.24·51-s − 1.09·53-s − 2.11·57-s + 1.04·59-s + 2.04·61-s + 0.503·63-s + 0.733·67-s − 2.40·69-s + 0.936·73-s + 1.80·79-s + 11/9·81-s + 1.09·83-s − 1.67·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9120400720\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9120400720\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 24 T + 288 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
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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
−9.435330880818362861898421988574, −9.403435773382798172488660615074, −8.977489032846631961492653431021, −8.310488556918893265567780529407, −7.970434541080482739222639776938, −7.54279911710181256328349125209, −6.99585954222057837778936537293, −6.74290309768572867376203147843, −6.73297676475925844216356226049, −5.79225364350964496423268846699, −5.19295934726610047977433885556, −5.15012851878892312761354036258, −4.88516503467094398560240983087, −4.46462991134290027235382930180, −3.62600451083269632399138749568, −3.23976030722810859147189041380, −2.26110268668030789283699228157, −2.22956301135360237128084143282, −1.21720050887824962521273557664, −0.42983496778959672616558086028,
0.42983496778959672616558086028, 1.21720050887824962521273557664, 2.22956301135360237128084143282, 2.26110268668030789283699228157, 3.23976030722810859147189041380, 3.62600451083269632399138749568, 4.46462991134290027235382930180, 4.88516503467094398560240983087, 5.15012851878892312761354036258, 5.19295934726610047977433885556, 5.79225364350964496423268846699, 6.73297676475925844216356226049, 6.74290309768572867376203147843, 6.99585954222057837778936537293, 7.54279911710181256328349125209, 7.970434541080482739222639776938, 8.310488556918893265567780529407, 8.977489032846631961492653431021, 9.403435773382798172488660615074, 9.435330880818362861898421988574
