| L(s) = 1 | + 5.36·2-s − 0.836·3-s + 20.7·4-s − 1.92·5-s − 4.48·6-s − 17.6·7-s + 68.6·8-s − 26.3·9-s − 10.3·10-s + 11·11-s − 17.3·12-s − 94.8·14-s + 1.61·15-s + 201.·16-s − 29.3·17-s − 141.·18-s + 78.8·19-s − 40.0·20-s + 14.7·21-s + 59.0·22-s − 167.·23-s − 57.3·24-s − 121.·25-s + 44.5·27-s − 367.·28-s − 191.·29-s + 8.65·30-s + ⋯ |
| L(s) = 1 | + 1.89·2-s − 0.160·3-s + 2.59·4-s − 0.172·5-s − 0.305·6-s − 0.954·7-s + 3.03·8-s − 0.974·9-s − 0.327·10-s + 0.301·11-s − 0.418·12-s − 1.81·14-s + 0.0277·15-s + 3.15·16-s − 0.418·17-s − 1.84·18-s + 0.951·19-s − 0.448·20-s + 0.153·21-s + 0.571·22-s − 1.52·23-s − 0.488·24-s − 0.970·25-s + 0.317·27-s − 2.47·28-s − 1.22·29-s + 0.0526·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 5.36T + 8T^{2} \) |
| 3 | \( 1 + 0.836T + 27T^{2} \) |
| 5 | \( 1 + 1.92T + 125T^{2} \) |
| 7 | \( 1 + 17.6T + 343T^{2} \) |
| 17 | \( 1 + 29.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 78.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 167.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 191.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 47.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 27.3T + 5.06e4T^{2} \) |
| 41 | \( 1 - 297.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 522.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 27.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + 179.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 92.1T + 2.05e5T^{2} \) |
| 61 | \( 1 - 824.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 224.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 461.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 883.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 150.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.13e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.35e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 328.T + 9.12e5T^{2} \) |
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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
−8.200353530844099806941955725843, −7.33367643889885026334498818437, −6.52338861440830983649752726210, −5.85946360735560665118303249660, −5.38710090488197426577408609117, −4.21401700143599875362625365908, −3.57888298932609160858051289513, −2.83684853387326663990502461231, −1.82866055556528886747143396182, 0,
1.82866055556528886747143396182, 2.83684853387326663990502461231, 3.57888298932609160858051289513, 4.21401700143599875362625365908, 5.38710090488197426577408609117, 5.85946360735560665118303249660, 6.52338861440830983649752726210, 7.33367643889885026334498818437, 8.200353530844099806941955725843
