| L(s) = 1 | + 2-s − 4-s − 5-s − 3·7-s − 3·8-s − 10-s − 2·11-s − 2·13-s − 3·14-s − 16-s + 4·17-s − 8·19-s + 20-s − 2·22-s + 3·23-s + 25-s − 2·26-s + 3·28-s − 29-s + 5·32-s + 4·34-s + 3·35-s − 4·37-s − 8·38-s + 3·40-s + 5·41-s − 8·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.447·5-s − 1.13·7-s − 1.06·8-s − 0.316·10-s − 0.603·11-s − 0.554·13-s − 0.801·14-s − 1/4·16-s + 0.970·17-s − 1.83·19-s + 0.223·20-s − 0.426·22-s + 0.625·23-s + 1/5·25-s − 0.392·26-s + 0.566·28-s − 0.185·29-s + 0.883·32-s + 0.685·34-s + 0.507·35-s − 0.657·37-s − 1.29·38-s + 0.474·40-s + 0.780·41-s − 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.77469071826085469350446623927, −9.898527011341948118823413441919, −9.019286204430249539319170001575, −8.053933674867411547003077582597, −6.84595624283791199980948871072, −5.86971236166707955375003452449, −4.82687436247101660115281422977, −3.77402379572268668385012510927, −2.80534791056945823123328757931, 0,
2.80534791056945823123328757931, 3.77402379572268668385012510927, 4.82687436247101660115281422977, 5.86971236166707955375003452449, 6.84595624283791199980948871072, 8.053933674867411547003077582597, 9.019286204430249539319170001575, 9.898527011341948118823413441919, 10.77469071826085469350446623927
