L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 3·7-s − 8-s + 9-s − 10-s + 11-s − 12-s − 13-s − 3·14-s − 15-s + 16-s − 17-s − 18-s − 5·19-s + 20-s − 3·21-s − 22-s + 5·23-s + 24-s + 25-s + 26-s − 27-s + 3·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s − 0.288·12-s − 0.277·13-s − 0.801·14-s − 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 1.14·19-s + 0.223·20-s − 0.654·21-s − 0.213·22-s + 1.04·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−7.88058149628121160902764474018, −7.03212124495856352666449083307, −6.50855774547019508889175165569, −5.66797487239975754462834163988, −4.94809421359120563933137306180, −4.29662797664554055937546700794, −3.07701758829316573831775380931, −1.93317415307731796669883552553, −1.41252040396299039554779366447, 0,
1.41252040396299039554779366447, 1.93317415307731796669883552553, 3.07701758829316573831775380931, 4.29662797664554055937546700794, 4.94809421359120563933137306180, 5.66797487239975754462834163988, 6.50855774547019508889175165569, 7.03212124495856352666449083307, 7.88058149628121160902764474018