| L(s) = 1 | + 0.783·2-s − 2.20·3-s − 7.38·4-s − 9.28·5-s − 1.72·6-s − 12.0·8-s − 22.1·9-s − 7.27·10-s + 25.2·11-s + 16.2·12-s + 3.79·13-s + 20.4·15-s + 49.6·16-s − 108.·17-s − 17.3·18-s − 60.1·19-s + 68.6·20-s + 19.7·22-s + 110.·23-s + 26.5·24-s − 38.7·25-s + 2.96·26-s + 108.·27-s + 11.9·29-s + 16.0·30-s + 122.·31-s + 135.·32-s + ⋯ |
| L(s) = 1 | + 0.276·2-s − 0.424·3-s − 0.923·4-s − 0.830·5-s − 0.117·6-s − 0.532·8-s − 0.820·9-s − 0.230·10-s + 0.692·11-s + 0.391·12-s + 0.0809·13-s + 0.352·15-s + 0.775·16-s − 1.54·17-s − 0.227·18-s − 0.725·19-s + 0.767·20-s + 0.191·22-s + 1.00·23-s + 0.225·24-s − 0.309·25-s + 0.0224·26-s + 0.772·27-s + 0.0764·29-s + 0.0975·30-s + 0.707·31-s + 0.747·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 47 | \( 1 + 47T \) |
| good | 2 | \( 1 - 0.783T + 8T^{2} \) |
| 3 | \( 1 + 2.20T + 27T^{2} \) |
| 5 | \( 1 + 9.28T + 125T^{2} \) |
| 11 | \( 1 - 25.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 3.79T + 2.19e3T^{2} \) |
| 17 | \( 1 + 108.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 60.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 110.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 11.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 122.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 314.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 138.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 220.T + 7.95e4T^{2} \) |
| 53 | \( 1 - 489.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 394.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 68.4T + 2.26e5T^{2} \) |
| 67 | \( 1 + 111.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 411.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 108.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 970.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.01e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 397.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.11e3T + 9.12e5T^{2} \) |
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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
−8.549609490853171828865825357855, −7.54491709076902763920448523061, −6.50881492815228323353751580706, −5.95152105139644526805252697608, −4.85298566142218301234411012207, −4.34796000314694004955298234710, −3.55912535312739815875795454736, −2.49646626418688172567387866711, −0.844915394710203777096678862639, 0,
0.844915394710203777096678862639, 2.49646626418688172567387866711, 3.55912535312739815875795454736, 4.34796000314694004955298234710, 4.85298566142218301234411012207, 5.95152105139644526805252697608, 6.50881492815228323353751580706, 7.54491709076902763920448523061, 8.549609490853171828865825357855
