L(s) = 1 | − 4.58·2-s + 13.0·4-s + 15.1·5-s − 25.8·7-s − 23.2·8-s − 69.4·10-s − 0.294·11-s + 3.35·13-s + 118.·14-s + 2.01·16-s − 39.1·17-s + 11.2·19-s + 197.·20-s + 1.35·22-s − 44.6·23-s + 103.·25-s − 15.3·26-s − 337.·28-s − 75.9·29-s − 34.1·31-s + 176.·32-s + 179.·34-s − 390.·35-s + 114.·37-s − 51.3·38-s − 350.·40-s + 364.·41-s + ⋯ |
L(s) = 1 | − 1.62·2-s + 1.63·4-s + 1.35·5-s − 1.39·7-s − 1.02·8-s − 2.19·10-s − 0.00808·11-s + 0.0715·13-s + 2.26·14-s + 0.0314·16-s − 0.558·17-s + 0.135·19-s + 2.20·20-s + 0.0131·22-s − 0.404·23-s + 0.830·25-s − 0.116·26-s − 2.27·28-s − 0.486·29-s − 0.197·31-s + 0.974·32-s + 0.906·34-s − 1.88·35-s + 0.509·37-s − 0.219·38-s − 1.38·40-s + 1.38·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 239 | \( 1 + 239T \) |
good | 2 | \( 1 + 4.58T + 8T^{2} \) |
| 5 | \( 1 - 15.1T + 125T^{2} \) |
| 7 | \( 1 + 25.8T + 343T^{2} \) |
| 11 | \( 1 + 0.294T + 1.33e3T^{2} \) |
| 13 | \( 1 - 3.35T + 2.19e3T^{2} \) |
| 17 | \( 1 + 39.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 11.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 44.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 75.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 34.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 114.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 364.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 145.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 322.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 345.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 701.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 357.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 748.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 730.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 41.7T + 3.89e5T^{2} \) |
| 79 | \( 1 - 450.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 826.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.29e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 533.T + 9.12e5T^{2} \) |
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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
−8.684114184353100606182954682704, −7.60807809515235387108996750985, −6.89952732464799639446294289103, −6.18280990643193031924318556282, −5.64094219799653069204810173569, −4.13421468852126607864079218227, −2.78372749459842987293565115744, −2.14936356974835570214024282739, −1.05657662754521387367535002743, 0,
1.05657662754521387367535002743, 2.14936356974835570214024282739, 2.78372749459842987293565115744, 4.13421468852126607864079218227, 5.64094219799653069204810173569, 6.18280990643193031924318556282, 6.89952732464799639446294289103, 7.60807809515235387108996750985, 8.684114184353100606182954682704