| L(s) = 1 | + 3.27·2-s + 3·3-s + 2.75·4-s − 17.5·5-s + 9.83·6-s + 26.6·7-s − 17.2·8-s + 9·9-s − 57.5·10-s + 21.4·11-s + 8.25·12-s + 87.5·14-s − 52.6·15-s − 78.4·16-s + 83.9·17-s + 29.5·18-s + 77.1·19-s − 48.2·20-s + 80.0·21-s + 70.1·22-s + 142.·23-s − 51.6·24-s + 182.·25-s + 27·27-s + 73.4·28-s + 134.·29-s − 172.·30-s + ⋯ |
| L(s) = 1 | + 1.15·2-s + 0.577·3-s + 0.343·4-s − 1.56·5-s + 0.669·6-s + 1.44·7-s − 0.760·8-s + 0.333·9-s − 1.81·10-s + 0.586·11-s + 0.198·12-s + 1.67·14-s − 0.905·15-s − 1.22·16-s + 1.19·17-s + 0.386·18-s + 0.931·19-s − 0.539·20-s + 0.832·21-s + 0.680·22-s + 1.28·23-s − 0.439·24-s + 1.46·25-s + 0.192·27-s + 0.495·28-s + 0.859·29-s − 1.05·30-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
| \(L(2)\) |
\(\approx\) |
\(3.834647157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.834647157\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 3.27T + 8T^{2} \) |
| 5 | \( 1 + 17.5T + 125T^{2} \) |
| 7 | \( 1 - 26.6T + 343T^{2} \) |
| 11 | \( 1 - 21.4T + 1.33e3T^{2} \) |
| 17 | \( 1 - 83.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 77.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 142.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 134.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 122.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 222.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 198.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 154.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 78.7T + 1.03e5T^{2} \) |
| 53 | \( 1 + 477.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 42.9T + 2.05e5T^{2} \) |
| 61 | \( 1 - 496.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 484.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 382.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 193.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.04e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 861.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 967.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 591.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.96964356491969755465590991173, −9.443005166316117004456262209250, −8.453876685385557427043891985789, −7.81617302664015745638978498672, −6.96190680147123611939608171732, −5.37424927976595706873872243976, −4.61248082207851100305963434479, −3.80848945753517248972996185345, −2.97788426530784169368829774308, −1.08325499369405240793353764882,
1.08325499369405240793353764882, 2.97788426530784169368829774308, 3.80848945753517248972996185345, 4.61248082207851100305963434479, 5.37424927976595706873872243976, 6.96190680147123611939608171732, 7.81617302664015745638978498672, 8.453876685385557427043891985789, 9.443005166316117004456262209250, 10.96964356491969755465590991173
