L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s + 7-s − 8-s + 9-s + 4·11-s + 2·12-s − 14-s + 16-s + 6·17-s − 18-s − 6·19-s + 2·21-s − 4·22-s − 23-s − 2·24-s − 5·25-s − 4·27-s + 28-s + 10·29-s + 4·31-s − 32-s + 8·33-s − 6·34-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 1.20·11-s + 0.577·12-s − 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 1.37·19-s + 0.436·21-s − 0.852·22-s − 0.208·23-s − 0.408·24-s − 25-s − 0.769·27-s + 0.188·28-s + 1.85·29-s + 0.718·31-s − 0.176·32-s + 1.39·33-s − 1.02·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.460221336\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.460221336\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−11.67721895089542655866728336195, −10.37016164816248582735310146228, −9.658483908820657946749564133485, −8.564938525169103336774729590767, −8.256405483321132934201523023612, −7.08283561954096782227932168001, −5.96045888935663136782137565226, −4.20633391117940512317281660230, −3.00348732867246236098130942847, −1.62045793899539357321283424119,
1.62045793899539357321283424119, 3.00348732867246236098130942847, 4.20633391117940512317281660230, 5.96045888935663136782137565226, 7.08283561954096782227932168001, 8.256405483321132934201523023612, 8.564938525169103336774729590767, 9.658483908820657946749564133485, 10.37016164816248582735310146228, 11.67721895089542655866728336195