L(s) = 1 | + 2-s − 2·3-s + 4-s + 5-s − 2·6-s + 7-s + 8-s + 9-s + 10-s − 2·11-s − 2·12-s − 4·13-s + 14-s − 2·15-s + 16-s − 2·17-s + 18-s − 4·19-s + 20-s − 2·21-s − 2·22-s + 23-s − 2·24-s + 25-s − 4·26-s + 4·27-s + 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.447·5-s − 0.816·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s − 0.577·12-s − 1.10·13-s + 0.267·14-s − 0.516·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.917·19-s + 0.223·20-s − 0.436·21-s − 0.426·22-s + 0.208·23-s − 0.408·24-s + 1/5·25-s − 0.784·26-s + 0.769·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−9.081318881140574859801927829423, −8.042301254362367544856865513618, −7.07410426121419441287557354290, −6.43247332212867595127782146298, −5.49565282564436954838683587718, −5.08262870085277344346932602417, −4.25441768705844202919228509076, −2.83550922629644707082968451602, −1.82145833143919711474940805671, 0,
1.82145833143919711474940805671, 2.83550922629644707082968451602, 4.25441768705844202919228509076, 5.08262870085277344346932602417, 5.49565282564436954838683587718, 6.43247332212867595127782146298, 7.07410426121419441287557354290, 8.042301254362367544856865513618, 9.081318881140574859801927829423