| 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
| \(L(1)\) |
\(\approx\) |
\(1.393494048\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.393494048\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 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
−10.16859351086247851025105791810, −9.640071709600070347331925838773, −9.576480149181028029468024761089, −9.271726308994816244280619594904, −8.706108431812127578304744353087, −8.159009584040943836754208767695, −7.52890313441364507043717555437, −7.08551808863849815405925565338, −6.66311011722927919707263303019, −6.58024055513476895465898157119, −5.77477784657855480082064813370, −5.76470860108653836015411529248, −4.77448897721646626211453931696, −4.66295310994620697106268586990, −3.68362201472377614093225082887, −3.54979302518516662793457994536, −2.87848973299083312316921824190, −2.82940459869884827336252932759, −1.52807265628850812338325120329, −0.50100639163660691663184068019,
0.50100639163660691663184068019, 1.52807265628850812338325120329, 2.82940459869884827336252932759, 2.87848973299083312316921824190, 3.54979302518516662793457994536, 3.68362201472377614093225082887, 4.66295310994620697106268586990, 4.77448897721646626211453931696, 5.76470860108653836015411529248, 5.77477784657855480082064813370, 6.58024055513476895465898157119, 6.66311011722927919707263303019, 7.08551808863849815405925565338, 7.52890313441364507043717555437, 8.159009584040943836754208767695, 8.706108431812127578304744353087, 9.271726308994816244280619594904, 9.576480149181028029468024761089, 9.640071709600070347331925838773, 10.16859351086247851025105791810
