| L(s) = 1 | − 2.34·2-s − 2.49·4-s − 15.3·5-s − 10.1·7-s + 24.6·8-s + 36.1·10-s − 15.0·11-s + 23.7·14-s − 37.8·16-s − 90.8·17-s + 114.·19-s + 38.3·20-s + 35.3·22-s − 75.7·23-s + 112.·25-s + 25.2·28-s − 214.·29-s − 284.·31-s − 108.·32-s + 213.·34-s + 156.·35-s + 358.·37-s − 269.·38-s − 379.·40-s − 313.·41-s − 296.·43-s + 37.5·44-s + ⋯ |
| L(s) = 1 | − 0.829·2-s − 0.311·4-s − 1.37·5-s − 0.547·7-s + 1.08·8-s + 1.14·10-s − 0.412·11-s + 0.453·14-s − 0.591·16-s − 1.29·17-s + 1.38·19-s + 0.428·20-s + 0.342·22-s − 0.686·23-s + 0.897·25-s + 0.170·28-s − 1.37·29-s − 1.64·31-s − 0.597·32-s + 1.07·34-s + 0.753·35-s + 1.59·37-s − 1.14·38-s − 1.49·40-s − 1.19·41-s − 1.05·43-s + 0.128·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.02532053394\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02532053394\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.34T + 8T^{2} \) |
| 5 | \( 1 + 15.3T + 125T^{2} \) |
| 7 | \( 1 + 10.1T + 343T^{2} \) |
| 11 | \( 1 + 15.0T + 1.33e3T^{2} \) |
| 17 | \( 1 + 90.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 114.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 75.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 214.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 284.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 358.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 313.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 296.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 316.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 163.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 254.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 935.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 240.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 947.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 430.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 496.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 392.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 979.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 553.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.115101774398162367372125076702, −8.316256976339539710107540531569, −7.53117892344209719399733234629, −7.20701634084670448379324825600, −5.86429091252433410263719644868, −4.75303835159913066173689576404, −4.00153208146597971362552853032, −3.14161628329464441088707436638, −1.61543002118959117918168624390, −0.088696848292400492926589752764,
0.088696848292400492926589752764, 1.61543002118959117918168624390, 3.14161628329464441088707436638, 4.00153208146597971362552853032, 4.75303835159913066173689576404, 5.86429091252433410263719644868, 7.20701634084670448379324825600, 7.53117892344209719399733234629, 8.316256976339539710107540531569, 9.115101774398162367372125076702
