| L(s) = 1 | + 3.87·2-s + 3.68·3-s + 7.02·4-s − 0.729·5-s + 14.2·6-s + 10.5·7-s − 3.78·8-s − 13.4·9-s − 2.82·10-s + 11·11-s + 25.8·12-s + 40.8·14-s − 2.68·15-s − 70.8·16-s − 3.76·17-s − 52.0·18-s − 74.4·19-s − 5.12·20-s + 38.8·21-s + 42.6·22-s − 121.·23-s − 13.9·24-s − 124.·25-s − 148.·27-s + 73.9·28-s − 121.·29-s − 10.4·30-s + ⋯ |
| L(s) = 1 | + 1.37·2-s + 0.709·3-s + 0.877·4-s − 0.0652·5-s + 0.971·6-s + 0.568·7-s − 0.167·8-s − 0.497·9-s − 0.0893·10-s + 0.301·11-s + 0.622·12-s + 0.779·14-s − 0.0462·15-s − 1.10·16-s − 0.0537·17-s − 0.681·18-s − 0.898·19-s − 0.0572·20-s + 0.403·21-s + 0.413·22-s − 1.10·23-s − 0.118·24-s − 0.995·25-s − 1.06·27-s + 0.499·28-s − 0.780·29-s − 0.0633·30-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 | \( 1 - 3.87T + 8T^{2} \) |
| 3 | \( 1 - 3.68T + 27T^{2} \) |
| 5 | \( 1 + 0.729T + 125T^{2} \) |
| 7 | \( 1 - 10.5T + 343T^{2} \) |
| 17 | \( 1 + 3.76T + 4.91e3T^{2} \) |
| 19 | \( 1 + 74.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 121.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 121.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 285.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 194.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 4.70T + 6.89e4T^{2} \) |
| 43 | \( 1 + 286.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 477.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 565.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 145.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 539.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 607.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 478.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 349.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.15e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 791.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 20.0T + 7.04e5T^{2} \) |
| 97 | \( 1 + 251.T + 9.12e5T^{2} \) |
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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
−8.295753157821401689471905853900, −7.86071746591831766983757873752, −6.55152229022228084179030842716, −6.00521188634287528817610765665, −5.07872125224128098432770453244, −4.24255919641939106515580395144, −3.61131995621622390936489566287, −2.63609972194617225680487326918, −1.84682952299145841743035485338, 0,
1.84682952299145841743035485338, 2.63609972194617225680487326918, 3.61131995621622390936489566287, 4.24255919641939106515580395144, 5.07872125224128098432770453244, 6.00521188634287528817610765665, 6.55152229022228084179030842716, 7.86071746591831766983757873752, 8.295753157821401689471905853900
