| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s − 4·7-s + 8-s + 9-s + 2·11-s − 2·12-s − 4·14-s + 16-s − 2·17-s + 18-s − 6·19-s + 8·21-s + 2·22-s − 6·23-s − 2·24-s + 4·27-s − 4·28-s + 2·29-s + 6·31-s + 32-s − 4·33-s − 2·34-s + 36-s − 2·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.603·11-s − 0.577·12-s − 1.06·14-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 1.37·19-s + 1.74·21-s + 0.426·22-s − 1.25·23-s − 0.408·24-s + 0.769·27-s − 0.755·28-s + 0.371·29-s + 1.07·31-s + 0.176·32-s − 0.696·33-s − 0.342·34-s + 1/6·36-s − 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8797014231\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8797014231\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−7.49717863670555268984235704948, −6.56732147635976301509796221164, −6.27116571092244297559237034494, −6.07633596781838763680895337796, −4.95174659211595008495472287932, −4.39221808182671573021746148699, −3.59609675393067732470185752555, −2.83822924379783627525508957408, −1.81314854697175844210266706717, −0.41778976177783236320310895724,
0.41778976177783236320310895724, 1.81314854697175844210266706717, 2.83822924379783627525508957408, 3.59609675393067732470185752555, 4.39221808182671573021746148699, 4.95174659211595008495472287932, 6.07633596781838763680895337796, 6.27116571092244297559237034494, 6.56732147635976301509796221164, 7.49717863670555268984235704948
