| L(s) = 1 | + 2-s − 3-s + 4-s + 2·5-s − 6-s + 8-s + 9-s + 2·10-s − 11-s − 12-s − 2·15-s + 16-s − 17-s + 18-s + 4·19-s + 2·20-s − 22-s − 4·23-s − 24-s − 25-s − 27-s − 6·29-s − 2·30-s − 8·31-s + 32-s + 33-s − 34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.301·11-s − 0.288·12-s − 0.516·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.917·19-s + 0.447·20-s − 0.213·22-s − 0.834·23-s − 0.204·24-s − 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.365·30-s − 1.43·31-s + 0.176·32-s + 0.174·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.298052858\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.298052858\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−12.99719477987004, −12.86786669334105, −12.17846728222970, −11.73033049124247, −11.30926387285466, −10.87580349123251, −10.24926253225278, −9.990796249990652, −9.411962753803005, −8.973229474403057, −8.306499004386698, −7.498581519147910, −7.396287114754685, −6.703580396603442, −6.121330120994443, −5.709889841965248, −5.414954094235975, −4.895801725711205, −4.265557592251490, −3.649191250226295, −3.197732710870459, −2.393476913301313, −1.798594198478083, −1.487710315830960, −0.3659255040165123,
0.3659255040165123, 1.487710315830960, 1.798594198478083, 2.393476913301313, 3.197732710870459, 3.649191250226295, 4.265557592251490, 4.895801725711205, 5.414954094235975, 5.709889841965248, 6.121330120994443, 6.703580396603442, 7.396287114754685, 7.498581519147910, 8.306499004386698, 8.973229474403057, 9.411962753803005, 9.990796249990652, 10.24926253225278, 10.87580349123251, 11.30926387285466, 11.73033049124247, 12.17846728222970, 12.86786669334105, 12.99719477987004
