L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 7-s − 8-s − 2·9-s + 10-s + 12-s + 7·13-s + 14-s − 15-s + 16-s + 6·17-s + 2·18-s − 5·19-s − 20-s − 21-s + 9·23-s − 24-s + 25-s − 7·26-s − 5·27-s − 28-s + 30-s + 2·31-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.316·10-s + 0.288·12-s + 1.94·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.471·18-s − 1.14·19-s − 0.223·20-s − 0.218·21-s + 1.87·23-s − 0.204·24-s + 1/5·25-s − 1.37·26-s − 0.962·27-s − 0.188·28-s + 0.182·30-s + 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.633856814\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.633856814\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−7.984395796404464441179408298843, −7.27809271178842750970068445575, −6.45400146142969656975681718873, −5.95316505843366481812114437640, −5.10059401455594828965140643877, −3.92334921649343197754349238514, −3.32341518671414085080257050749, −2.80312173087626467788631323231, −1.58546820293200595927931731830, −0.70623039461998375672077263189,
0.70623039461998375672077263189, 1.58546820293200595927931731830, 2.80312173087626467788631323231, 3.32341518671414085080257050749, 3.92334921649343197754349238514, 5.10059401455594828965140643877, 5.95316505843366481812114437640, 6.45400146142969656975681718873, 7.27809271178842750970068445575, 7.984395796404464441179408298843