| L(s) = 1 | − 2-s + 3-s + 4-s + 3.46·5-s − 6-s + 7-s − 8-s + 9-s − 3.46·10-s + 11-s + 12-s + 2·13-s − 14-s + 3.46·15-s + 16-s − 3.46·17-s − 18-s − 1.46·19-s + 3.46·20-s + 21-s − 22-s − 6.92·23-s − 24-s + 6.99·25-s − 2·26-s + 27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.54·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 1.09·10-s + 0.301·11-s + 0.288·12-s + 0.554·13-s − 0.267·14-s + 0.894·15-s + 0.250·16-s − 0.840·17-s − 0.235·18-s − 0.335·19-s + 0.774·20-s + 0.218·21-s − 0.213·22-s − 1.44·23-s − 0.204·24-s + 1.39·25-s − 0.392·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.643760195\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.643760195\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 3.46T + 5T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 1.46T + 19T^{2} \) |
| 23 | \( 1 + 6.92T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 - 8.92T + 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 - 2.92T + 43T^{2} \) |
| 47 | \( 1 - 2.53T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 + 6.92T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 1.07T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 7.46T + 73T^{2} \) |
| 79 | \( 1 - 2.92T + 79T^{2} \) |
| 83 | \( 1 + 16.3T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−10.74433393642895475367766421449, −10.00110105167259080413415084245, −9.214714422376758870179865567047, −8.617718601682383826649694374982, −7.53358436179697876158945181235, −6.39419680149113395802921236634, −5.67761341162773234140283236806, −4.11932092109000370232963063510, −2.46076100486846570220703570998, −1.62497531587841803918805722140,
1.62497531587841803918805722140, 2.46076100486846570220703570998, 4.11932092109000370232963063510, 5.67761341162773234140283236806, 6.39419680149113395802921236634, 7.53358436179697876158945181235, 8.617718601682383826649694374982, 9.214714422376758870179865567047, 10.00110105167259080413415084245, 10.74433393642895475367766421449
