| L(s) = 1 | − 2-s − 3-s + 4-s + 3.27·5-s + 6-s + 7-s − 8-s + 9-s − 3.27·10-s + 5.27·11-s − 12-s − 13-s − 14-s − 3.27·15-s + 16-s − 7.27·17-s − 18-s + 5.27·19-s + 3.27·20-s − 21-s − 5.27·22-s − 5.27·23-s + 24-s + 5.72·25-s + 26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.46·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 1.03·10-s + 1.59·11-s − 0.288·12-s − 0.277·13-s − 0.267·14-s − 0.845·15-s + 0.250·16-s − 1.76·17-s − 0.235·18-s + 1.21·19-s + 0.732·20-s − 0.218·21-s − 1.12·22-s − 1.09·23-s + 0.204·24-s + 1.14·25-s + 0.196·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.237634902\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.237634902\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 3.27T + 5T^{2} \) |
| 11 | \( 1 - 5.27T + 11T^{2} \) |
| 17 | \( 1 + 7.27T + 17T^{2} \) |
| 19 | \( 1 - 5.27T + 19T^{2} \) |
| 23 | \( 1 + 5.27T + 23T^{2} \) |
| 29 | \( 1 - 0.725T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 3.27T + 37T^{2} \) |
| 41 | \( 1 + 8.54T + 41T^{2} \) |
| 43 | \( 1 - 5.27T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 4.72T + 61T^{2} \) |
| 67 | \( 1 + 2.54T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 9.82T + 73T^{2} \) |
| 79 | \( 1 + 2.54T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 + 15.0T + 97T^{2} \) |
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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
−10.66146478055000204937840010278, −9.767772604161815803693236117002, −9.309884670475511043909031425770, −8.343518506082386869702031908164, −6.91026514947932260405007550027, −6.39870794343943751494465686215, −5.47193319690133672490048174894, −4.24062375293323472675955780464, −2.35289732891516897669516912027, −1.28606116306891868803211902621,
1.28606116306891868803211902621, 2.35289732891516897669516912027, 4.24062375293323472675955780464, 5.47193319690133672490048174894, 6.39870794343943751494465686215, 6.91026514947932260405007550027, 8.343518506082386869702031908164, 9.309884670475511043909031425770, 9.767772604161815803693236117002, 10.66146478055000204937840010278
