| L(s) = 1 | − 1.42·2-s + 3-s + 0.0397·4-s − 0.0606·5-s − 1.42·6-s + 1.70·7-s + 2.79·8-s + 9-s + 0.0866·10-s − 11-s + 0.0397·12-s − 2.43·14-s − 0.0606·15-s − 4.07·16-s + 3.75·17-s − 1.42·18-s − 2.02·19-s − 0.00241·20-s + 1.70·21-s + 1.42·22-s + 0.704·23-s + 2.79·24-s − 4.99·25-s + 27-s + 0.0678·28-s − 0.0346·29-s + 0.0866·30-s + ⋯ |
| L(s) = 1 | − 1.00·2-s + 0.577·3-s + 0.0198·4-s − 0.0271·5-s − 0.583·6-s + 0.644·7-s + 0.989·8-s + 0.333·9-s + 0.0273·10-s − 0.301·11-s + 0.0114·12-s − 0.651·14-s − 0.0156·15-s − 1.01·16-s + 0.910·17-s − 0.336·18-s − 0.464·19-s − 0.000538·20-s + 0.372·21-s + 0.304·22-s + 0.146·23-s + 0.571·24-s − 0.999·25-s + 0.192·27-s + 0.0128·28-s − 0.00644·29-s + 0.0158·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.42T + 2T^{2} \) |
| 5 | \( 1 + 0.0606T + 5T^{2} \) |
| 7 | \( 1 - 1.70T + 7T^{2} \) |
| 17 | \( 1 - 3.75T + 17T^{2} \) |
| 19 | \( 1 + 2.02T + 19T^{2} \) |
| 23 | \( 1 - 0.704T + 23T^{2} \) |
| 29 | \( 1 + 0.0346T + 29T^{2} \) |
| 31 | \( 1 + 1.85T + 31T^{2} \) |
| 37 | \( 1 + 8.82T + 37T^{2} \) |
| 41 | \( 1 + 3.32T + 41T^{2} \) |
| 43 | \( 1 - 5.29T + 43T^{2} \) |
| 47 | \( 1 + 6.04T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 + 3.34T + 59T^{2} \) |
| 61 | \( 1 - 3.26T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 + 1.96T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 + 4.19T + 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 + 5.86T + 97T^{2} \) |
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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
−7.972684937048684139325263354332, −7.46348123573505276336988588087, −6.66008298076374251530167959208, −5.54870451478890364758155833889, −4.85200668267170690719809146522, −4.04863030248324384072391837104, −3.18652240596425331539302509698, −2.00420351263148829246891068586, −1.37045985446278621408605905177, 0,
1.37045985446278621408605905177, 2.00420351263148829246891068586, 3.18652240596425331539302509698, 4.04863030248324384072391837104, 4.85200668267170690719809146522, 5.54870451478890364758155833889, 6.66008298076374251530167959208, 7.46348123573505276336988588087, 7.972684937048684139325263354332
