| L(s) = 1 | − 3·3-s − 2·5-s + 3·7-s + 4·9-s + 7·11-s − 3·13-s + 6·15-s + 3·17-s − 19-s − 9·21-s − 2·23-s + 3·25-s − 6·27-s − 2·29-s − 5·31-s − 21·33-s − 6·35-s − 16·37-s + 9·39-s − 9·41-s + 4·43-s − 8·45-s + 2·47-s − 4·49-s − 9·51-s + 8·53-s − 14·55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.894·5-s + 1.13·7-s + 4/3·9-s + 2.11·11-s − 0.832·13-s + 1.54·15-s + 0.727·17-s − 0.229·19-s − 1.96·21-s − 0.417·23-s + 3/5·25-s − 1.15·27-s − 0.371·29-s − 0.898·31-s − 3.65·33-s − 1.01·35-s − 2.63·37-s + 1.44·39-s − 1.40·41-s + 0.609·43-s − 1.19·45-s + 0.291·47-s − 4/7·49-s − 1.26·51-s + 1.09·53-s − 1.88·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54169600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54169600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_4$ | \( 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 3 T + 13 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 7 T + 31 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 25 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 7 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 46 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 5 T + 39 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 9 T + 73 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 82 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5 T + 47 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 29 T + 349 T^{2} + 29 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 10 T + 54 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 130 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 9 T + 211 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
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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
−7.50566752367698501639112375208, −7.18584322510690673768686935598, −7.14667009495707518574667036505, −6.72236717090708343800605199924, −6.32509308048312561714142557207, −5.96002464026647663803941422649, −5.48865571271591317563562964026, −5.35342812878046297551027278564, −4.94001319653524944707158482513, −4.68186054890195905046869141343, −4.01259858580927532142217898112, −3.99542928913818942937158401881, −3.55971082638454320585024823335, −3.13634902836226910176117881554, −2.24340247467640058739608131715, −1.84257208463999278687447181318, −1.35479781565662687111880634573, −1.04293745896348535530588738323, 0, 0,
1.04293745896348535530588738323, 1.35479781565662687111880634573, 1.84257208463999278687447181318, 2.24340247467640058739608131715, 3.13634902836226910176117881554, 3.55971082638454320585024823335, 3.99542928913818942937158401881, 4.01259858580927532142217898112, 4.68186054890195905046869141343, 4.94001319653524944707158482513, 5.35342812878046297551027278564, 5.48865571271591317563562964026, 5.96002464026647663803941422649, 6.32509308048312561714142557207, 6.72236717090708343800605199924, 7.14667009495707518574667036505, 7.18584322510690673768686935598, 7.50566752367698501639112375208
