L(s) = 1 | + 2-s + 3-s + 4-s − 2.73·5-s + 6-s + 7-s + 8-s + 9-s − 2.73·10-s + 5.19·11-s + 12-s + 14-s − 2.73·15-s + 16-s − 5.73·17-s + 18-s − 6.46·19-s − 2.73·20-s + 21-s + 5.19·22-s − 0.535·23-s + 24-s + 2.46·25-s + 27-s + 28-s − 10.4·29-s − 2.73·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.22·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.863·10-s + 1.56·11-s + 0.288·12-s + 0.267·14-s − 0.705·15-s + 0.250·16-s − 1.39·17-s + 0.235·18-s − 1.48·19-s − 0.610·20-s + 0.218·21-s + 1.10·22-s − 0.111·23-s + 0.204·24-s + 0.492·25-s + 0.192·27-s + 0.188·28-s − 1.94·29-s − 0.498·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 5 | \( 1 + 2.73T + 5T^{2} \) |
| 11 | \( 1 - 5.19T + 11T^{2} \) |
| 17 | \( 1 + 5.73T + 17T^{2} \) |
| 19 | \( 1 + 6.46T + 19T^{2} \) |
| 23 | \( 1 + 0.535T + 23T^{2} \) |
| 29 | \( 1 + 10.4T + 29T^{2} \) |
| 31 | \( 1 + 6.73T + 31T^{2} \) |
| 37 | \( 1 + 3.26T + 37T^{2} \) |
| 41 | \( 1 + 1.53T + 41T^{2} \) |
| 43 | \( 1 + 5.26T + 43T^{2} \) |
| 47 | \( 1 - 9.92T + 47T^{2} \) |
| 53 | \( 1 - 3.92T + 53T^{2} \) |
| 59 | \( 1 + 4.92T + 59T^{2} \) |
| 61 | \( 1 - 0.267T + 61T^{2} \) |
| 67 | \( 1 - 10T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 + 5.46T + 73T^{2} \) |
| 79 | \( 1 - 9T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 3.80T + 97T^{2} \) |
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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
−7.47503987585896035305854796694, −6.92103059479008605283675970639, −6.33846598943127450334182278048, −5.33989595158785071607480605357, −4.30072528008946937512558008115, −4.02534180532585803122346388191, −3.54117225658809382692224821416, −2.28227700541208189107535460452, −1.62024100904737084568858725009, 0,
1.62024100904737084568858725009, 2.28227700541208189107535460452, 3.54117225658809382692224821416, 4.02534180532585803122346388191, 4.30072528008946937512558008115, 5.33989595158785071607480605357, 6.33846598943127450334182278048, 6.92103059479008605283675970639, 7.47503987585896035305854796694