L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 2.61·7-s − 8-s + 9-s − 10-s − 1.41·11-s − 12-s − 1.23·13-s − 2.61·14-s − 15-s + 16-s − 18-s − 2.82·19-s + 20-s − 2.61·21-s + 1.41·22-s + 0.0630·23-s + 24-s + 25-s + 1.23·26-s − 27-s + 2.61·28-s − 2.75·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 0.987·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.426·11-s − 0.288·12-s − 0.342·13-s − 0.698·14-s − 0.258·15-s + 0.250·16-s − 0.235·18-s − 0.648·19-s + 0.223·20-s − 0.570·21-s + 0.301·22-s + 0.0131·23-s + 0.204·24-s + 0.200·25-s + 0.242·26-s − 0.192·27-s + 0.493·28-s − 0.510·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 7 | \( 1 - 2.61T + 7T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 13 | \( 1 + 1.23T + 13T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 0.0630T + 23T^{2} \) |
| 29 | \( 1 + 2.75T + 29T^{2} \) |
| 31 | \( 1 + 1.75T + 31T^{2} \) |
| 37 | \( 1 + 3.06T + 37T^{2} \) |
| 41 | \( 1 - 5.24T + 41T^{2} \) |
| 43 | \( 1 - 6.58T + 43T^{2} \) |
| 47 | \( 1 + 1.06T + 47T^{2} \) |
| 53 | \( 1 - 4.52T + 53T^{2} \) |
| 59 | \( 1 + 2.05T + 59T^{2} \) |
| 61 | \( 1 - 0.304T + 61T^{2} \) |
| 67 | \( 1 - 9.94T + 67T^{2} \) |
| 71 | \( 1 + 7.83T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 - 7.27T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 + 15.0T + 89T^{2} \) |
| 97 | \( 1 - 2.21T + 97T^{2} \) |
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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
−7.41155449288773142906201250950, −6.91714378862285498191469528725, −5.97303238741353946384541509321, −5.50711636406605828797245886159, −4.74573014484723667908851110606, −4.00192113082076327207842980715, −2.75645150150947331992398706622, −1.99857381207146225118795411417, −1.21611972278022431151364382862, 0,
1.21611972278022431151364382862, 1.99857381207146225118795411417, 2.75645150150947331992398706622, 4.00192113082076327207842980715, 4.74573014484723667908851110606, 5.50711636406605828797245886159, 5.97303238741353946384541509321, 6.91714378862285498191469528725, 7.41155449288773142906201250950