| L(s) = 1 | − 2.25·2-s + 4.02·3-s − 2.93·4-s + 4.98·5-s − 9.06·6-s + 5.25·7-s + 24.6·8-s − 10.7·9-s − 11.2·10-s + 25.9·11-s − 11.8·12-s + 14.2·13-s − 11.8·14-s + 20.0·15-s − 31.9·16-s + 20.3·17-s + 24.2·18-s − 2.21·19-s − 14.6·20-s + 21.1·21-s − 58.4·22-s + 39.1·23-s + 99.1·24-s − 100.·25-s − 31.9·26-s − 152.·27-s − 15.3·28-s + ⋯ |
| L(s) = 1 | − 0.795·2-s + 0.775·3-s − 0.366·4-s + 0.446·5-s − 0.616·6-s + 0.283·7-s + 1.08·8-s − 0.399·9-s − 0.355·10-s + 0.711·11-s − 0.284·12-s + 0.303·13-s − 0.225·14-s + 0.345·15-s − 0.499·16-s + 0.290·17-s + 0.317·18-s − 0.0267·19-s − 0.163·20-s + 0.219·21-s − 0.566·22-s + 0.354·23-s + 0.843·24-s − 0.800·25-s − 0.241·26-s − 1.08·27-s − 0.103·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 \) |
| good | 2 | \( 1 + 2.25T + 8T^{2} \) |
| 3 | \( 1 - 4.02T + 27T^{2} \) |
| 5 | \( 1 - 4.98T + 125T^{2} \) |
| 7 | \( 1 - 5.25T + 343T^{2} \) |
| 11 | \( 1 - 25.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 14.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 20.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 2.21T + 6.85e3T^{2} \) |
| 23 | \( 1 - 39.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 277.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 47.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 43.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 98.7T + 6.89e4T^{2} \) |
| 47 | \( 1 - 488.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 230.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 400.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 13.3T + 2.26e5T^{2} \) |
| 67 | \( 1 + 640.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 251.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 526.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 978.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 37.2T + 5.71e5T^{2} \) |
| 89 | \( 1 + 437.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.12e3T + 9.12e5T^{2} \) |
| 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
−8.632292596035223393551144710134, −7.894474099844757804827408942834, −7.31568717091844830668396688658, −6.08960312152215454543734886045, −5.31456260766652001982968752001, −4.17360406701294702057668202018, −3.41837594665037695884902418735, −2.12746331479052446594116746422, −1.33480025013575151007468640415, 0,
1.33480025013575151007468640415, 2.12746331479052446594116746422, 3.41837594665037695884902418735, 4.17360406701294702057668202018, 5.31456260766652001982968752001, 6.08960312152215454543734886045, 7.31568717091844830668396688658, 7.894474099844757804827408942834, 8.632292596035223393551144710134
