| L(s) = 1 | + 5.46·2-s − 5.57·3-s + 21.8·4-s + 6.77·5-s − 30.5·6-s + 4.72·7-s + 75.9·8-s + 4.13·9-s + 37.0·10-s + 9.47·11-s − 122.·12-s + 33.7·13-s + 25.8·14-s − 37.8·15-s + 240.·16-s + 22.5·18-s + 27.4·19-s + 148.·20-s − 26.3·21-s + 51.8·22-s − 82.8·23-s − 423.·24-s − 79.0·25-s + 184.·26-s + 127.·27-s + 103.·28-s + 208.·29-s + ⋯ |
| L(s) = 1 | + 1.93·2-s − 1.07·3-s + 2.73·4-s + 0.605·5-s − 2.07·6-s + 0.254·7-s + 3.35·8-s + 0.153·9-s + 1.17·10-s + 0.259·11-s − 2.93·12-s + 0.719·13-s + 0.492·14-s − 0.650·15-s + 3.75·16-s + 0.295·18-s + 0.331·19-s + 1.65·20-s − 0.273·21-s + 0.502·22-s − 0.751·23-s − 3.60·24-s − 0.632·25-s + 1.39·26-s + 0.909·27-s + 0.697·28-s + 1.33·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.102417636\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.102417636\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 - 5.46T + 8T^{2} \) |
| 3 | \( 1 + 5.57T + 27T^{2} \) |
| 5 | \( 1 - 6.77T + 125T^{2} \) |
| 7 | \( 1 - 4.72T + 343T^{2} \) |
| 11 | \( 1 - 9.47T + 1.33e3T^{2} \) |
| 13 | \( 1 - 33.7T + 2.19e3T^{2} \) |
| 19 | \( 1 - 27.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 82.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 208.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 223.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 173.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 86.1T + 6.89e4T^{2} \) |
| 43 | \( 1 + 258.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 88.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + 541.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 13.5T + 2.05e5T^{2} \) |
| 61 | \( 1 + 158.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 357.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 960.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 898.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 451.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 559.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 602.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 820.T + 9.12e5T^{2} \) |
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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
−11.80073936187268364907933948389, −10.88319007027518406985963914480, −10.07619311087261484125427920974, −8.132304892935769831010380105869, −6.62588913065068775511829936993, −6.16152079138811271713846708470, −5.29001864175223985100235613049, −4.44187290082541877858081965005, −3.08333628037575779492326225252, −1.56021608252201509983284368977,
1.56021608252201509983284368977, 3.08333628037575779492326225252, 4.44187290082541877858081965005, 5.29001864175223985100235613049, 6.16152079138811271713846708470, 6.62588913065068775511829936993, 8.132304892935769831010380105869, 10.07619311087261484125427920974, 10.88319007027518406985963914480, 11.80073936187268364907933948389
