| L(s) = 1 | − 3-s + 2·5-s − 7-s − 9-s + 2·11-s − 2·13-s − 2·15-s − 4·17-s + 7·19-s + 21-s + 3·25-s − 6·29-s − 4·31-s − 2·33-s − 2·35-s + 2·39-s − 10·41-s − 4·43-s − 2·45-s − 6·47-s − 9·49-s + 4·51-s + 4·53-s + 4·55-s − 7·57-s − 18·59-s + 10·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.377·7-s − 1/3·9-s + 0.603·11-s − 0.554·13-s − 0.516·15-s − 0.970·17-s + 1.60·19-s + 0.218·21-s + 3/5·25-s − 1.11·29-s − 0.718·31-s − 0.348·33-s − 0.338·35-s + 0.320·39-s − 1.56·41-s − 0.609·43-s − 0.298·45-s − 0.875·47-s − 9/7·49-s + 0.560·51-s + 0.549·53-s + 0.539·55-s − 0.927·57-s − 2.34·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36481600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36481600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 151 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_4$ | \( 1 - 7 T + 46 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 29 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 49 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 10 T + 90 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 86 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 10 T + 130 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 2 T + 126 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2 T + 79 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 10 T + 123 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 22 T + 282 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 130 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.82100797118544324570425740448, −7.42661560122772483318352343471, −7.03905546617237501079366043988, −6.92013750737318921353389405276, −6.31365565082444495505778107206, −6.24530192991694714520287515472, −5.70689408493793104722804547425, −5.42971363086014532713759134556, −5.10927441128920111702496012987, −4.79862232065495656381656140721, −4.33442611046960920993616809952, −3.78369945725601525781474695597, −3.30360802228921267599748524781, −3.14746091517404342107096244161, −2.50384709666552456333115737033, −2.03965610050917304495721583321, −1.52346674114436636520465419574, −1.21436722948717851252929564920, 0, 0,
1.21436722948717851252929564920, 1.52346674114436636520465419574, 2.03965610050917304495721583321, 2.50384709666552456333115737033, 3.14746091517404342107096244161, 3.30360802228921267599748524781, 3.78369945725601525781474695597, 4.33442611046960920993616809952, 4.79862232065495656381656140721, 5.10927441128920111702496012987, 5.42971363086014532713759134556, 5.70689408493793104722804547425, 6.24530192991694714520287515472, 6.31365565082444495505778107206, 6.92013750737318921353389405276, 7.03905546617237501079366043988, 7.42661560122772483318352343471, 7.82100797118544324570425740448
