| L(s) = 1 | − 2-s + 2·3-s + 4-s + 5-s − 2·6-s − 4·7-s − 8-s + 9-s − 10-s − 2·11-s + 2·12-s + 4·14-s + 2·15-s + 16-s + 6·17-s − 18-s − 19-s + 20-s − 8·21-s + 2·22-s + 9·23-s − 2·24-s − 4·25-s − 4·27-s − 4·28-s − 9·29-s − 2·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.447·5-s − 0.816·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.603·11-s + 0.577·12-s + 1.06·14-s + 0.516·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.229·19-s + 0.223·20-s − 1.74·21-s + 0.426·22-s + 1.87·23-s − 0.408·24-s − 4/5·25-s − 0.769·27-s − 0.755·28-s − 1.67·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.664733230\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.664733230\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−7.70950855891956151934606498259, −7.44191220619433966018961367930, −6.68647975931380937882543472176, −5.69089616095608260073551089780, −5.41702278465834633014763152330, −3.68491109867412852064776117626, −3.45843147022291223690386547402, −2.60364062511111766377917888309, −1.97254502671739664822329900695, −0.64064263439333474493415276610,
0.64064263439333474493415276610, 1.97254502671739664822329900695, 2.60364062511111766377917888309, 3.45843147022291223690386547402, 3.68491109867412852064776117626, 5.41702278465834633014763152330, 5.69089616095608260073551089780, 6.68647975931380937882543472176, 7.44191220619433966018961367930, 7.70950855891956151934606498259
