L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s − 2·11-s − 12-s + 2·13-s − 14-s + 15-s + 16-s − 18-s + 6·19-s − 20-s − 21-s + 2·22-s + 6·23-s + 24-s + 25-s − 2·26-s − 27-s + 28-s − 2·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s − 0.288·12-s + 0.554·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s + 1.37·19-s − 0.223·20-s − 0.218·21-s + 0.426·22-s + 1.25·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.188·28-s − 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.371796559\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.371796559\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−14.16082848395625, −13.93907639555207, −13.14544810615121, −12.72066749422204, −12.08172754849878, −11.70875249696671, −11.07476249585108, −10.89851145689632, −10.25669262120018, −9.777442087997459, −9.020259555213877, −8.735215657522524, −8.043864016128661, −7.469375476121975, −7.149800373310867, −6.591016661787588, −5.727271794934419, −5.355819775336168, −4.851497197597086, −3.986932916689002, −3.384992334376370, −2.739084097978386, −1.881243781320964, −1.099637840739877, −0.5463712388585088,
0.5463712388585088, 1.099637840739877, 1.881243781320964, 2.739084097978386, 3.384992334376370, 3.986932916689002, 4.851497197597086, 5.355819775336168, 5.727271794934419, 6.591016661787588, 7.149800373310867, 7.469375476121975, 8.043864016128661, 8.735215657522524, 9.020259555213877, 9.777442087997459, 10.25669262120018, 10.89851145689632, 11.07476249585108, 11.70875249696671, 12.08172754849878, 12.72066749422204, 13.14544810615121, 13.93907639555207, 14.16082848395625