| L(s) = 1 | − 2-s + 0.496·3-s + 4-s + 5-s − 0.496·6-s + 0.154·7-s − 8-s − 2.75·9-s − 10-s − 11-s + 0.496·12-s + 3.41·13-s − 0.154·14-s + 0.496·15-s + 16-s + 0.773·17-s + 2.75·18-s + 6.71·19-s + 20-s + 0.0767·21-s + 22-s − 6.55·23-s − 0.496·24-s + 25-s − 3.41·26-s − 2.85·27-s + 0.154·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.286·3-s + 0.5·4-s + 0.447·5-s − 0.202·6-s + 0.0583·7-s − 0.353·8-s − 0.917·9-s − 0.316·10-s − 0.301·11-s + 0.143·12-s + 0.947·13-s − 0.0412·14-s + 0.128·15-s + 0.250·16-s + 0.187·17-s + 0.648·18-s + 1.54·19-s + 0.223·20-s + 0.0167·21-s + 0.213·22-s − 1.36·23-s − 0.101·24-s + 0.200·25-s − 0.669·26-s − 0.549·27-s + 0.0291·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| good | 3 | \( 1 - 0.496T + 3T^{2} \) |
| 7 | \( 1 - 0.154T + 7T^{2} \) |
| 13 | \( 1 - 3.41T + 13T^{2} \) |
| 17 | \( 1 - 0.773T + 17T^{2} \) |
| 19 | \( 1 - 6.71T + 19T^{2} \) |
| 23 | \( 1 + 6.55T + 23T^{2} \) |
| 29 | \( 1 - 8.44T + 29T^{2} \) |
| 31 | \( 1 + 5.89T + 31T^{2} \) |
| 37 | \( 1 + 4.82T + 37T^{2} \) |
| 41 | \( 1 + 12.3T + 41T^{2} \) |
| 43 | \( 1 + 8.81T + 43T^{2} \) |
| 47 | \( 1 - 6.60T + 47T^{2} \) |
| 53 | \( 1 + 3.65T + 53T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 + 5.52T + 61T^{2} \) |
| 67 | \( 1 - 1.64T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 79 | \( 1 + 7.30T + 79T^{2} \) |
| 83 | \( 1 - 3.90T + 83T^{2} \) |
| 89 | \( 1 - 0.0565T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.70024268347585206182390089680, −6.83536149530311109617344120183, −6.15187549470604523282477990189, −5.54267766479464861300602170064, −4.85038173137698333835757506571, −3.46758551253509801381466547919, −3.16087372670995341664637335131, −2.06723838568243500925784825217, −1.31242738740780204691487145447, 0,
1.31242738740780204691487145447, 2.06723838568243500925784825217, 3.16087372670995341664637335131, 3.46758551253509801381466547919, 4.85038173137698333835757506571, 5.54267766479464861300602170064, 6.15187549470604523282477990189, 6.83536149530311109617344120183, 7.70024268347585206182390089680
