| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 3.48·7-s + 8-s + 9-s + 10-s + 3.32·11-s − 12-s + 4.58·13-s + 3.48·14-s − 15-s + 16-s + 18-s + 5.23·19-s + 20-s − 3.48·21-s + 3.32·22-s + 2.85·23-s − 24-s + 25-s + 4.58·26-s − 27-s + 3.48·28-s − 4.08·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s + 1.31·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 1.00·11-s − 0.288·12-s + 1.27·13-s + 0.932·14-s − 0.258·15-s + 0.250·16-s + 0.235·18-s + 1.20·19-s + 0.223·20-s − 0.761·21-s + 0.708·22-s + 0.595·23-s − 0.204·24-s + 0.200·25-s + 0.900·26-s − 0.192·27-s + 0.659·28-s − 0.758·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.671783657\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.671783657\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 - 3.48T + 7T^{2} \) |
| 11 | \( 1 - 3.32T + 11T^{2} \) |
| 13 | \( 1 - 4.58T + 13T^{2} \) |
| 19 | \( 1 - 5.23T + 19T^{2} \) |
| 23 | \( 1 - 2.85T + 23T^{2} \) |
| 29 | \( 1 + 4.08T + 29T^{2} \) |
| 31 | \( 1 + 6.29T + 31T^{2} \) |
| 37 | \( 1 - 8.93T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 - 2.30T + 43T^{2} \) |
| 47 | \( 1 + 2.23T + 47T^{2} \) |
| 53 | \( 1 + 5.09T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 + 7.38T + 61T^{2} \) |
| 67 | \( 1 + 2.27T + 67T^{2} \) |
| 71 | \( 1 + 1.25T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 + 3.85T + 83T^{2} \) |
| 89 | \( 1 - 7.99T + 89T^{2} \) |
| 97 | \( 1 + 6.79T + 97T^{2} \) |
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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
−7.55915091346770764992956533149, −7.03697160179145484609250698470, −6.06897741671942750607754247380, −5.75997707555808722899932650471, −5.04899671269645292130333493262, −4.29050855171675884526518754669, −3.73526172634653560322652407051, −2.68295772480896095012130129734, −1.48668041028611886445499215720, −1.19737881797973079514057620991,
1.19737881797973079514057620991, 1.48668041028611886445499215720, 2.68295772480896095012130129734, 3.73526172634653560322652407051, 4.29050855171675884526518754669, 5.04899671269645292130333493262, 5.75997707555808722899932650471, 6.06897741671942750607754247380, 7.03697160179145484609250698470, 7.55915091346770764992956533149
