| L(s) = 1 | − 2·3-s − 5·5-s + 2·7-s + 3·9-s + 10·11-s + 13-s + 10·15-s − 4·17-s − 4·19-s − 4·21-s − 2·23-s + 12·25-s − 4·27-s − 2·29-s − 6·31-s − 20·33-s − 10·35-s − 18·37-s − 2·39-s − 10·41-s + 7·43-s − 15·45-s + 10·47-s + 3·49-s + 8·51-s − 7·53-s − 50·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 2.23·5-s + 0.755·7-s + 9-s + 3.01·11-s + 0.277·13-s + 2.58·15-s − 0.970·17-s − 0.917·19-s − 0.872·21-s − 0.417·23-s + 12/5·25-s − 0.769·27-s − 0.371·29-s − 1.07·31-s − 3.48·33-s − 1.69·35-s − 2.95·37-s − 0.320·39-s − 1.56·41-s + 1.06·43-s − 2.23·45-s + 1.45·47-s + 3/7·49-s + 1.12·51-s − 0.961·53-s − 6.74·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59721984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59721984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + p T + 13 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - T + 23 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 25 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 29 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 10 T + 55 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 7 T + 17 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 10 T + 106 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 7 T + 37 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 3 T + 91 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 13 T + 135 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 13 T + 147 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 9 T + 133 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 10 T + 119 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 157 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 11 T + 179 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 28 T + 377 T^{2} + 28 p T^{3} + p^{2} T^{4} \) |
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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
−7.42735501432499161314849484389, −7.38077454501806412080360764699, −6.99938847239455516893047106437, −6.62619166709045898763086378010, −6.37265764925350221102434827378, −6.23569500465333154267857762879, −5.38145833087559759782643624451, −5.32874608905710848336060931194, −4.66190410374602781720437464886, −4.49266501787099364812786350392, −3.97139063985975143547150197410, −3.91889671583570099263211873335, −3.56730753411106306693623741931, −3.36276885191000819959010666380, −2.14052998086005200911494611372, −1.97143991476861053671338435353, −1.29986386827620440851268795892, −1.00012505549975965217365163366, 0, 0,
1.00012505549975965217365163366, 1.29986386827620440851268795892, 1.97143991476861053671338435353, 2.14052998086005200911494611372, 3.36276885191000819959010666380, 3.56730753411106306693623741931, 3.91889671583570099263211873335, 3.97139063985975143547150197410, 4.49266501787099364812786350392, 4.66190410374602781720437464886, 5.32874608905710848336060931194, 5.38145833087559759782643624451, 6.23569500465333154267857762879, 6.37265764925350221102434827378, 6.62619166709045898763086378010, 6.99938847239455516893047106437, 7.38077454501806412080360764699, 7.42735501432499161314849484389
