| L(s) = 1 | − 2-s + 8·7-s + 8-s + 3·9-s − 2·11-s − 2·13-s − 8·14-s − 16-s + 3·17-s − 3·18-s + 8·19-s + 2·22-s − 4·23-s + 2·26-s + 6·29-s − 4·31-s − 3·34-s − 4·37-s − 8·38-s + 3·41-s + 12·43-s + 4·46-s + 6·47-s + 34·49-s + 4·53-s + 8·56-s − 6·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 3.02·7-s + 0.353·8-s + 9-s − 0.603·11-s − 0.554·13-s − 2.13·14-s − 1/4·16-s + 0.727·17-s − 0.707·18-s + 1.83·19-s + 0.426·22-s − 0.834·23-s + 0.392·26-s + 1.11·29-s − 0.718·31-s − 0.514·34-s − 0.657·37-s − 1.29·38-s + 0.468·41-s + 1.82·43-s + 0.589·46-s + 0.875·47-s + 34/7·49-s + 0.549·53-s + 1.06·56-s − 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.508289287\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.508289287\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 12 T + 101 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 12 T + 83 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 15 T + 158 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 14 T + 117 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - T - 88 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−10.38450045593392763058529638412, −9.806945868444657179890078391459, −9.331278319024421238205381058932, −9.131648018292945956548558624444, −8.304064873109822080349928709952, −8.160092209846945261948423177638, −7.74449496139154088767916504511, −7.59955754260840875186267303582, −7.16683350197101172488231284634, −6.64712956023782217981691171034, −5.58134872982081990061053675284, −5.37910592099521989355386862677, −5.13721948123108281567837603195, −4.48585311121576836448253852871, −4.21259459093084646413143670001, −3.55849083310874862777676318559, −2.47815361197550296732342454157, −2.13962698116482825132393866715, −1.21933920243924547826322587540, −1.09921874520690091350756836785,
1.09921874520690091350756836785, 1.21933920243924547826322587540, 2.13962698116482825132393866715, 2.47815361197550296732342454157, 3.55849083310874862777676318559, 4.21259459093084646413143670001, 4.48585311121576836448253852871, 5.13721948123108281567837603195, 5.37910592099521989355386862677, 5.58134872982081990061053675284, 6.64712956023782217981691171034, 7.16683350197101172488231284634, 7.59955754260840875186267303582, 7.74449496139154088767916504511, 8.160092209846945261948423177638, 8.304064873109822080349928709952, 9.131648018292945956548558624444, 9.331278319024421238205381058932, 9.806945868444657179890078391459, 10.38450045593392763058529638412
