| L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s + 11-s − 6·13-s + 14-s + 16-s + 6·17-s − 4·19-s + 20-s − 22-s + 4·23-s + 25-s + 6·26-s − 28-s − 2·29-s − 32-s − 6·34-s − 35-s + 2·37-s + 4·38-s − 40-s + 2·41-s − 4·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.301·11-s − 1.66·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.917·19-s + 0.223·20-s − 0.213·22-s + 0.834·23-s + 1/5·25-s + 1.17·26-s − 0.188·28-s − 0.371·29-s − 0.176·32-s − 1.02·34-s − 0.169·35-s + 0.328·37-s + 0.648·38-s − 0.158·40-s + 0.312·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.250044086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.250044086\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−7.83658762461569849895473846235, −7.37276950628160052064074417217, −6.69320815478101761806124137842, −5.94242070947557982230294896987, −5.25024410935228830221741977129, −4.42856090469910489800582607495, −3.32422490525917516841296562229, −2.61354523930823816520784745014, −1.76143917309672845507967559786, −0.62821807979168704674965459748,
0.62821807979168704674965459748, 1.76143917309672845507967559786, 2.61354523930823816520784745014, 3.32422490525917516841296562229, 4.42856090469910489800582607495, 5.25024410935228830221741977129, 5.94242070947557982230294896987, 6.69320815478101761806124137842, 7.37276950628160052064074417217, 7.83658762461569849895473846235
