| L(s) = 1 | + 2.39·2-s + 3·3-s − 2.24·4-s + 1.37·5-s + 7.19·6-s − 24.5·8-s + 9·9-s + 3.30·10-s − 11·11-s − 6.74·12-s + 25.1·13-s + 4.12·15-s − 40.9·16-s − 44.6·17-s + 21.5·18-s + 108.·19-s − 3.09·20-s − 26.3·22-s − 78.8·23-s − 73.7·24-s − 123.·25-s + 60.4·26-s + 27·27-s + 259.·29-s + 9.90·30-s + 108.·31-s + 98.3·32-s + ⋯ |
| L(s) = 1 | + 0.847·2-s + 0.577·3-s − 0.280·4-s + 0.123·5-s + 0.489·6-s − 1.08·8-s + 0.333·9-s + 0.104·10-s − 0.301·11-s − 0.162·12-s + 0.537·13-s + 0.0710·15-s − 0.640·16-s − 0.636·17-s + 0.282·18-s + 1.30·19-s − 0.0345·20-s − 0.255·22-s − 0.714·23-s − 0.627·24-s − 0.984·25-s + 0.455·26-s + 0.192·27-s + 1.66·29-s + 0.0602·30-s + 0.627·31-s + 0.543·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 2.39T + 8T^{2} \) |
| 5 | \( 1 - 1.37T + 125T^{2} \) |
| 13 | \( 1 - 25.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 44.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 108.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 78.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 259.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 108.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 349.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 234.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 15.1T + 7.95e4T^{2} \) |
| 47 | \( 1 + 288.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 703.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 903.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 283.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 252.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 875.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 773.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 135.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 194.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 929.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.54e3T + 9.12e5T^{2} \) |
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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
−8.500464189577439476317642320140, −8.059175064888434922096712888275, −6.85050694533803224063632078649, −6.08933231937519974276179615368, −5.16000441500566611227091275129, −4.44467467082171177544465351412, −3.49301967849191310581645445229, −2.83309064549925591467198583446, −1.52755503227984962321556201650, 0,
1.52755503227984962321556201650, 2.83309064549925591467198583446, 3.49301967849191310581645445229, 4.44467467082171177544465351412, 5.16000441500566611227091275129, 6.08933231937519974276179615368, 6.85050694533803224063632078649, 8.059175064888434922096712888275, 8.500464189577439476317642320140
