| L(s) = 1 | + 5.33·2-s − 3·3-s + 20.4·4-s − 16.4·5-s − 15.9·6-s + 9.67·7-s + 66.1·8-s + 9·9-s − 87.4·10-s + 27.5·11-s − 61.2·12-s + 51.5·14-s + 49.2·15-s + 189.·16-s + 107.·17-s + 47.9·18-s − 2.24·19-s − 335.·20-s − 29.0·21-s + 147.·22-s + 41.8·23-s − 198.·24-s + 144.·25-s − 27·27-s + 197.·28-s + 61.6·29-s + 262.·30-s + ⋯ |
| L(s) = 1 | + 1.88·2-s − 0.577·3-s + 2.55·4-s − 1.46·5-s − 1.08·6-s + 0.522·7-s + 2.92·8-s + 0.333·9-s − 2.76·10-s + 0.756·11-s − 1.47·12-s + 0.984·14-s + 0.847·15-s + 2.95·16-s + 1.53·17-s + 0.628·18-s − 0.0271·19-s − 3.74·20-s − 0.301·21-s + 1.42·22-s + 0.379·23-s − 1.68·24-s + 1.15·25-s − 0.192·27-s + 1.33·28-s + 0.394·29-s + 1.59·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.018003499\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.018003499\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 5.33T + 8T^{2} \) |
| 5 | \( 1 + 16.4T + 125T^{2} \) |
| 7 | \( 1 - 9.67T + 343T^{2} \) |
| 11 | \( 1 - 27.5T + 1.33e3T^{2} \) |
| 17 | \( 1 - 107.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 2.24T + 6.85e3T^{2} \) |
| 23 | \( 1 - 41.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 61.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 191.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 98.4T + 5.06e4T^{2} \) |
| 41 | \( 1 + 30.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 238.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 511.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 492.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 484.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 444.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 190.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 484.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 957.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 375.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 715.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.03e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 65.5T + 9.12e5T^{2} \) |
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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
−11.18232734276806843516285050309, −10.08944398646647379528960555614, −8.233569800268697833388238331895, −7.43068865615885545887922617519, −6.60787715301137312614119853314, −5.56732827630202180957638223558, −4.63921703343560313295261108335, −3.96320384448111355598204922050, −3.02888905625715394873765922517, −1.20033374013883790220063816438,
1.20033374013883790220063816438, 3.02888905625715394873765922517, 3.96320384448111355598204922050, 4.63921703343560313295261108335, 5.56732827630202180957638223558, 6.60787715301137312614119853314, 7.43068865615885545887922617519, 8.233569800268697833388238331895, 10.08944398646647379528960555614, 11.18232734276806843516285050309
