| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 4·7-s + 8-s + 9-s − 10-s + 6·11-s − 12-s − 2·13-s + 4·14-s + 15-s + 16-s + 18-s − 19-s − 20-s − 4·21-s + 6·22-s + 8·23-s − 24-s + 25-s − 2·26-s − 27-s + 4·28-s + 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.80·11-s − 0.288·12-s − 0.554·13-s + 1.06·14-s + 0.258·15-s + 1/4·16-s + 0.235·18-s − 0.229·19-s − 0.223·20-s − 0.872·21-s + 1.27·22-s + 1.66·23-s − 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.755·28-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.771470156\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.771470156\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.18978575028607, −12.63931140934318, −12.16840463287743, −11.92201598865525, −11.29959933941281, −11.03890844208969, −10.87811987669820, −9.917637338584265, −9.468397395313569, −8.843241703213602, −8.479449469066130, −7.662998712903778, −7.443623173589855, −6.858317879703057, −6.319058002891007, −5.918579190667375, −5.074790054787399, −4.830696320997703, −4.361282576763322, −3.926132865357296, −3.232592340557210, −2.520938661859715, −1.761520486291928, −1.231018722738469, −0.7110771530288156,
0.7110771530288156, 1.231018722738469, 1.761520486291928, 2.520938661859715, 3.232592340557210, 3.926132865357296, 4.361282576763322, 4.830696320997703, 5.074790054787399, 5.918579190667375, 6.319058002891007, 6.858317879703057, 7.443623173589855, 7.662998712903778, 8.479449469066130, 8.843241703213602, 9.468397395313569, 9.917637338584265, 10.87811987669820, 11.03890844208969, 11.29959933941281, 11.92201598865525, 12.16840463287743, 12.63931140934318, 13.18978575028607
