| L(s) = 1 | + 2·3-s − 2·5-s + 7-s + 3·9-s − 3·11-s − 5·13-s − 4·15-s − 3·17-s + 19-s + 2·21-s − 3·23-s + 3·25-s + 4·27-s + 3·29-s − 5·31-s − 6·33-s − 2·35-s + 7·37-s − 10·39-s − 9·41-s − 8·43-s − 6·45-s + 3·47-s + 7·49-s − 6·51-s − 12·53-s + 6·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 0.377·7-s + 9-s − 0.904·11-s − 1.38·13-s − 1.03·15-s − 0.727·17-s + 0.229·19-s + 0.436·21-s − 0.625·23-s + 3/5·25-s + 0.769·27-s + 0.557·29-s − 0.898·31-s − 1.04·33-s − 0.338·35-s + 1.15·37-s − 1.60·39-s − 1.40·41-s − 1.21·43-s − 0.894·45-s + 0.437·47-s + 49-s − 0.840·51-s − 1.64·53-s + 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16160400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16160400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 67 | $C_2$ | \( 1 + 16 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 3 T - 62 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 17 T + 192 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
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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
−8.067172311913799644398522518899, −8.053009398975252598951507919156, −7.53609979189000858449249496835, −7.40808690279722983292084707155, −6.87688047559798852466584787865, −6.74536946460797627791957320597, −5.89667737031181707541371212422, −5.77833326632509560543595591078, −5.01419559473379600833467184140, −4.76420415923392057797651200723, −4.42520992245717355234626174129, −4.19217596240283791760325536946, −3.46584461151188074148618958531, −3.15904109307615042658096502722, −2.67081687129210301770040154055, −2.48700542010702159211854081058, −1.66762510054185849584593109074, −1.43006330164588165222317541853, 0, 0,
1.43006330164588165222317541853, 1.66762510054185849584593109074, 2.48700542010702159211854081058, 2.67081687129210301770040154055, 3.15904109307615042658096502722, 3.46584461151188074148618958531, 4.19217596240283791760325536946, 4.42520992245717355234626174129, 4.76420415923392057797651200723, 5.01419559473379600833467184140, 5.77833326632509560543595591078, 5.89667737031181707541371212422, 6.74536946460797627791957320597, 6.87688047559798852466584787865, 7.40808690279722983292084707155, 7.53609979189000858449249496835, 8.053009398975252598951507919156, 8.067172311913799644398522518899
