| L(s) = 1 | − 2.79·2-s + 10.1·3-s − 0.187·4-s − 5.45·5-s − 28.3·6-s − 7.73·7-s + 22.8·8-s + 75.9·9-s + 15.2·10-s + 11·11-s − 1.90·12-s + 21.6·14-s − 55.3·15-s − 62.4·16-s − 18.5·17-s − 212.·18-s − 65.8·19-s + 1.02·20-s − 78.4·21-s − 30.7·22-s + 38.8·23-s + 232.·24-s − 95.2·25-s + 496.·27-s + 1.45·28-s + 16.1·29-s + 154.·30-s + ⋯ |
| L(s) = 1 | − 0.988·2-s + 1.95·3-s − 0.0234·4-s − 0.488·5-s − 1.92·6-s − 0.417·7-s + 1.01·8-s + 2.81·9-s + 0.482·10-s + 0.301·11-s − 0.0458·12-s + 0.412·14-s − 0.953·15-s − 0.975·16-s − 0.264·17-s − 2.77·18-s − 0.794·19-s + 0.0114·20-s − 0.815·21-s − 0.297·22-s + 0.352·23-s + 1.97·24-s − 0.761·25-s + 3.53·27-s + 0.00980·28-s + 0.103·29-s + 0.941·30-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 | \( 1 + 2.79T + 8T^{2} \) |
| 3 | \( 1 - 10.1T + 27T^{2} \) |
| 5 | \( 1 + 5.45T + 125T^{2} \) |
| 7 | \( 1 + 7.73T + 343T^{2} \) |
| 17 | \( 1 + 18.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 65.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 38.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 16.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 100.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 327.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 460.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 346.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 318.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 387.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 93.9T + 2.05e5T^{2} \) |
| 61 | \( 1 - 801.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 344.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 186.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 820.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 502.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 199.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 535.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 462.T + 9.12e5T^{2} \) |
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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
−8.410487439630872992681825833368, −8.168195954897980793711750410541, −7.22058468355988769659508860840, −6.71084473497724641184700781246, −4.90245608298390751741001567772, −3.97144665629734059598829959385, −3.43121213393853276593705908445, −2.23197086278055038961084420703, −1.43566487817096994539414100186, 0,
1.43566487817096994539414100186, 2.23197086278055038961084420703, 3.43121213393853276593705908445, 3.97144665629734059598829959385, 4.90245608298390751741001567772, 6.71084473497724641184700781246, 7.22058468355988769659508860840, 8.168195954897980793711750410541, 8.410487439630872992681825833368
