| L(s) = 1 | − 2-s + 4-s − 5-s + 4·7-s − 8-s + 10-s − 2·11-s − 13-s − 4·14-s + 16-s + 6·19-s − 20-s + 2·22-s + 6·23-s + 25-s + 26-s + 4·28-s + 2·29-s + 6·31-s − 32-s − 4·35-s + 2·37-s − 6·38-s + 40-s + 10·41-s − 10·43-s − 2·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s − 0.353·8-s + 0.316·10-s − 0.603·11-s − 0.277·13-s − 1.06·14-s + 1/4·16-s + 1.37·19-s − 0.223·20-s + 0.426·22-s + 1.25·23-s + 1/5·25-s + 0.196·26-s + 0.755·28-s + 0.371·29-s + 1.07·31-s − 0.176·32-s − 0.676·35-s + 0.328·37-s − 0.973·38-s + 0.158·40-s + 1.56·41-s − 1.52·43-s − 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.475251217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.475251217\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−12.31428553116302, −11.92492201268380, −11.74346613859510, −11.09827443621922, −10.69317951301863, −10.57746880982889, −9.737672931543167, −9.373343324633453, −8.910367353139103, −8.320309252509789, −7.999964249836999, −7.599139464486610, −7.268006969516623, −6.737340209108056, −6.034901596372335, −5.437880719036577, −5.080493057869114, −4.544079277785941, −4.168134665205943, −3.188931839520367, −2.876913637464587, −2.316379532068483, −1.509035180116162, −1.101583118726196, −0.5215180710797357,
0.5215180710797357, 1.101583118726196, 1.509035180116162, 2.316379532068483, 2.876913637464587, 3.188931839520367, 4.168134665205943, 4.544079277785941, 5.080493057869114, 5.437880719036577, 6.034901596372335, 6.737340209108056, 7.268006969516623, 7.599139464486610, 7.999964249836999, 8.320309252509789, 8.910367353139103, 9.373343324633453, 9.737672931543167, 10.57746880982889, 10.69317951301863, 11.09827443621922, 11.74346613859510, 11.92492201268380, 12.31428553116302
