| L(s) = 1 | − 4·3-s + 2·5-s + 4·7-s + 8·9-s − 6·13-s − 8·15-s − 6·17-s − 16·21-s − 12·23-s − 25-s − 12·27-s + 8·35-s − 6·37-s + 24·39-s − 12·43-s + 16·45-s − 12·47-s + 8·49-s + 24·51-s − 6·53-s − 16·59-s − 12·61-s + 32·63-s − 12·65-s + 12·67-s + 48·69-s + 10·73-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 0.894·5-s + 1.51·7-s + 8/3·9-s − 1.66·13-s − 2.06·15-s − 1.45·17-s − 3.49·21-s − 2.50·23-s − 1/5·25-s − 2.30·27-s + 1.35·35-s − 0.986·37-s + 3.84·39-s − 1.82·43-s + 2.38·45-s − 1.75·47-s + 8/7·49-s + 3.36·51-s − 0.824·53-s − 2.08·59-s − 1.53·61-s + 4.03·63-s − 1.48·65-s + 1.46·67-s + 5.77·69-s + 1.17·73-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
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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
−9.510068305118607769667969282961, −9.408887756577383664907424207566, −8.581502404159138738954688101602, −8.060088137276290538705919340885, −7.908996560457538008532018614127, −7.30227835306332592970253743319, −6.78014348839458342586090814749, −6.40657561447329141454833232388, −6.14587873090926796891878724028, −5.61326405579647884905062035838, −5.24383173031467372298135042947, −4.96278191408211701351926304571, −4.49158944607474995317860323441, −4.30475816928882914416401626860, −3.33950870353528692256487602720, −2.33522688916145357185460140694, −1.74207533632421601016936417025, −1.68375498005259898135911888355, 0, 0,
1.68375498005259898135911888355, 1.74207533632421601016936417025, 2.33522688916145357185460140694, 3.33950870353528692256487602720, 4.30475816928882914416401626860, 4.49158944607474995317860323441, 4.96278191408211701351926304571, 5.24383173031467372298135042947, 5.61326405579647884905062035838, 6.14587873090926796891878724028, 6.40657561447329141454833232388, 6.78014348839458342586090814749, 7.30227835306332592970253743319, 7.908996560457538008532018614127, 8.060088137276290538705919340885, 8.581502404159138738954688101602, 9.408887756577383664907424207566, 9.510068305118607769667969282961
