| L(s) = 1 | − 2·5-s + 6·13-s − 10·17-s − 7·25-s − 10·29-s − 10·37-s + 4·41-s + 10·49-s − 4·53-s − 26·61-s − 12·65-s + 6·73-s + 20·85-s − 26·89-s − 12·97-s − 20·101-s − 18·109-s − 2·113-s + 2·121-s + 26·125-s + 127-s + 131-s + 137-s + 139-s + 20·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.66·13-s − 2.42·17-s − 7/5·25-s − 1.85·29-s − 1.64·37-s + 0.624·41-s + 10/7·49-s − 0.549·53-s − 3.32·61-s − 1.48·65-s + 0.702·73-s + 2.16·85-s − 2.75·89-s − 1.21·97-s − 1.99·101-s − 1.72·109-s − 0.188·113-s + 2/11·121-s + 2.32·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.66·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 118 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−8.752319759274373161311462773208, −8.362937653540424186111951539314, −7.900127475677755310118239302558, −7.61125333803576586827498476869, −7.21907255126803469717129023630, −6.75035250370432786310001011093, −6.43449531214915693197831539216, −6.03964430557292507596160947072, −5.54997072464873009100009475867, −5.31404560489571684357332779442, −4.47475659220464340028771310734, −4.22586106229223050916277727219, −3.89955525618093693522040969850, −3.62137752344738492980747789716, −2.94539391068010149108226059733, −2.39327268126173847533542778342, −1.72925906536824499152018065956, −1.40804108458245971944468667846, 0, 0,
1.40804108458245971944468667846, 1.72925906536824499152018065956, 2.39327268126173847533542778342, 2.94539391068010149108226059733, 3.62137752344738492980747789716, 3.89955525618093693522040969850, 4.22586106229223050916277727219, 4.47475659220464340028771310734, 5.31404560489571684357332779442, 5.54997072464873009100009475867, 6.03964430557292507596160947072, 6.43449531214915693197831539216, 6.75035250370432786310001011093, 7.21907255126803469717129023630, 7.61125333803576586827498476869, 7.900127475677755310118239302558, 8.362937653540424186111951539314, 8.752319759274373161311462773208
