| L(s) = 1 | − 4.68·2-s − 2.64·3-s + 13.9·4-s − 16.8·5-s + 12.3·6-s − 13.2·7-s − 28.0·8-s − 20.0·9-s + 79.2·10-s − 11·11-s − 36.9·12-s + 62.3·14-s + 44.6·15-s + 19.6·16-s − 38.1·17-s + 93.8·18-s − 48.0·19-s − 236.·20-s + 35.1·21-s + 51.5·22-s − 161.·23-s + 74.1·24-s + 160.·25-s + 124.·27-s − 185.·28-s + 303.·29-s − 209.·30-s + ⋯ |
| L(s) = 1 | − 1.65·2-s − 0.508·3-s + 1.74·4-s − 1.51·5-s + 0.842·6-s − 0.717·7-s − 1.23·8-s − 0.741·9-s + 2.50·10-s − 0.301·11-s − 0.888·12-s + 1.18·14-s + 0.768·15-s + 0.307·16-s − 0.544·17-s + 1.22·18-s − 0.579·19-s − 2.64·20-s + 0.364·21-s + 0.499·22-s − 1.46·23-s + 0.630·24-s + 1.28·25-s + 0.885·27-s − 1.25·28-s + 1.94·29-s − 1.27·30-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 | \( 1 + 4.68T + 8T^{2} \) |
| 3 | \( 1 + 2.64T + 27T^{2} \) |
| 5 | \( 1 + 16.8T + 125T^{2} \) |
| 7 | \( 1 + 13.2T + 343T^{2} \) |
| 17 | \( 1 + 38.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 48.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 161.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 303.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 261.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 405.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 241.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 184.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 80.8T + 1.03e5T^{2} \) |
| 53 | \( 1 - 472.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 260.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 336.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 417.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 140.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 496.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 120.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 561.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 607.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 49.5T + 9.12e5T^{2} \) |
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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
−8.293341729528764606632691769648, −8.147251570099646170852320142856, −6.93971870827145390028593656980, −6.64383510032891576337090054076, −5.43449856542786442036287346918, −4.21683965392705263648634953212, −3.24097917025196583994852518986, −2.12858640506250261780818760002, −0.56363589350013418841133830134, 0,
0.56363589350013418841133830134, 2.12858640506250261780818760002, 3.24097917025196583994852518986, 4.21683965392705263648634953212, 5.43449856542786442036287346918, 6.64383510032891576337090054076, 6.93971870827145390028593656980, 8.147251570099646170852320142856, 8.293341729528764606632691769648
