| L(s) = 1 | + 2-s + 3·3-s + 4·5-s + 3·6-s − 5·7-s − 8-s + 3·9-s + 4·10-s + 11-s − 2·13-s − 5·14-s + 12·15-s − 16-s − 2·17-s + 3·18-s − 6·19-s − 15·21-s + 22-s + 2·23-s − 3·24-s + 5·25-s − 2·26-s + 2·29-s + 12·30-s − 4·31-s + 3·33-s − 2·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s + 1.78·5-s + 1.22·6-s − 1.88·7-s − 0.353·8-s + 9-s + 1.26·10-s + 0.301·11-s − 0.554·13-s − 1.33·14-s + 3.09·15-s − 1/4·16-s − 0.485·17-s + 0.707·18-s − 1.37·19-s − 3.27·21-s + 0.213·22-s + 0.417·23-s − 0.612·24-s + 25-s − 0.392·26-s + 0.371·29-s + 2.19·30-s − 0.718·31-s + 0.522·33-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23716 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23716 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.734738175\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.734738175\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 9 T + 14 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 15 T + 146 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
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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
−13.38846371560432672660405971099, −12.96528287315446072102700425506, −12.51574387827071077619440490772, −12.20541195663546005208323134745, −11.04623671965480280605363191019, −10.55333383450835637937244143279, −9.735594596058672393343829737667, −9.664363201484915324423128282110, −9.286266462495648396467298288701, −8.617941701912037415993615121931, −8.406253260525527358347099467565, −7.17433214611602209685784023249, −6.82500268444453572766359887593, −6.01246055865774076949142190835, −5.89569493697794276035855828275, −4.85949936813717926247574447694, −3.90545201001924950217398459583, −3.35822139059151646624000002223, −2.39971980369470025644422909823, −2.35960317471923201559308793654,
2.35960317471923201559308793654, 2.39971980369470025644422909823, 3.35822139059151646624000002223, 3.90545201001924950217398459583, 4.85949936813717926247574447694, 5.89569493697794276035855828275, 6.01246055865774076949142190835, 6.82500268444453572766359887593, 7.17433214611602209685784023249, 8.406253260525527358347099467565, 8.617941701912037415993615121931, 9.286266462495648396467298288701, 9.664363201484915324423128282110, 9.735594596058672393343829737667, 10.55333383450835637937244143279, 11.04623671965480280605363191019, 12.20541195663546005208323134745, 12.51574387827071077619440490772, 12.96528287315446072102700425506, 13.38846371560432672660405971099
