L(s) = 1 | − 2-s − 2·3-s + 4-s − 2·5-s + 2·6-s − 8-s + 9-s + 2·10-s + 4·11-s − 2·12-s + 2·13-s + 4·15-s + 16-s + 4·17-s − 18-s − 2·19-s − 2·20-s − 4·22-s + 2·24-s − 25-s − 2·26-s + 4·27-s − 29-s − 4·30-s + 2·31-s − 32-s − 8·33-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.894·5-s + 0.816·6-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 1.20·11-s − 0.577·12-s + 0.554·13-s + 1.03·15-s + 1/4·16-s + 0.970·17-s − 0.235·18-s − 0.458·19-s − 0.447·20-s − 0.852·22-s + 0.408·24-s − 1/5·25-s − 0.392·26-s + 0.769·27-s − 0.185·29-s − 0.730·30-s + 0.359·31-s − 0.176·32-s − 1.39·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6111056509\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6111056509\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 29 | \( 1 + T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−8.605789181209095450571393447634, −8.181156406370071373473225265699, −7.19077389091838852362366810243, −6.54334625454827283506383172144, −5.92954536041548618947010825248, −5.00946084477045769784252885528, −4.00056290715485073203301368850, −3.24313368198724016020456507067, −1.61885400056636754162386078150, −0.59303062997997118440756794829,
0.59303062997997118440756794829, 1.61885400056636754162386078150, 3.24313368198724016020456507067, 4.00056290715485073203301368850, 5.00946084477045769784252885528, 5.92954536041548618947010825248, 6.54334625454827283506383172144, 7.19077389091838852362366810243, 8.181156406370071373473225265699, 8.605789181209095450571393447634