| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 2·7-s + 8-s + 9-s + 12-s + 6·13-s + 2·14-s + 16-s + 17-s + 18-s + 4·19-s + 2·21-s − 6·23-s + 24-s + 6·26-s + 27-s + 2·28-s − 4·29-s − 6·31-s + 32-s + 34-s + 36-s + 4·37-s + 4·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.288·12-s + 1.66·13-s + 0.534·14-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.917·19-s + 0.436·21-s − 1.25·23-s + 0.204·24-s + 1.17·26-s + 0.192·27-s + 0.377·28-s − 0.742·29-s − 1.07·31-s + 0.176·32-s + 0.171·34-s + 1/6·36-s + 0.657·37-s + 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.228729959\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.228729959\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−8.734828598079436554718045729045, −8.071448234457966112995261162971, −7.47986232645249780760032352510, −6.48622192953254051806046820765, −5.70560690897165992102208310924, −4.96302360112208612746504467278, −3.84262958907150087471004300345, −3.48996298828141413803939468236, −2.16754331116336168477722723471, −1.31372321402716153030334173496,
1.31372321402716153030334173496, 2.16754331116336168477722723471, 3.48996298828141413803939468236, 3.84262958907150087471004300345, 4.96302360112208612746504467278, 5.70560690897165992102208310924, 6.48622192953254051806046820765, 7.47986232645249780760032352510, 8.071448234457966112995261162971, 8.734828598079436554718045729045
