| L(s) = 1 | + 2-s + 4-s + 8-s − 4·11-s − 13-s + 16-s + 6·19-s − 4·22-s − 26-s − 6·29-s + 10·31-s + 32-s + 8·37-s + 6·38-s + 10·41-s − 2·43-s − 4·44-s − 6·47-s − 52-s + 2·53-s − 6·58-s + 10·62-s + 64-s + 12·67-s − 6·73-s + 8·74-s + 6·76-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 1.20·11-s − 0.277·13-s + 1/4·16-s + 1.37·19-s − 0.852·22-s − 0.196·26-s − 1.11·29-s + 1.79·31-s + 0.176·32-s + 1.31·37-s + 0.973·38-s + 1.56·41-s − 0.304·43-s − 0.603·44-s − 0.875·47-s − 0.138·52-s + 0.274·53-s − 0.787·58-s + 1.27·62-s + 1/8·64-s + 1.46·67-s − 0.702·73-s + 0.929·74-s + 0.688·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.461881800\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.461881800\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.77595532734018, −12.35981766456354, −11.80724101261930, −11.43821442397560, −10.98202254821394, −10.55910630879179, −9.884566824257494, −9.669093637380831, −9.188138998871527, −8.307763677359191, −7.998723846017557, −7.588633120809989, −7.149701719418696, −6.530704634801056, −6.009780578189657, −5.533513741381789, −5.117098671844916, −4.646305144558507, −4.123797767909244, −3.477379985145372, −2.902274208414858, −2.561303624213302, −1.963882550282352, −1.096075027754079, −0.5363244797785161,
0.5363244797785161, 1.096075027754079, 1.963882550282352, 2.561303624213302, 2.902274208414858, 3.477379985145372, 4.123797767909244, 4.646305144558507, 5.117098671844916, 5.533513741381789, 6.009780578189657, 6.530704634801056, 7.149701719418696, 7.588633120809989, 7.998723846017557, 8.307763677359191, 9.188138998871527, 9.669093637380831, 9.884566824257494, 10.55910630879179, 10.98202254821394, 11.43821442397560, 11.80724101261930, 12.35981766456354, 12.77595532734018
