| L(s) = 1 | + 3.53·2-s − 7.81·3-s + 4.49·4-s − 1.07·5-s − 27.6·6-s + 33.3·7-s − 12.3·8-s + 34.1·9-s − 3.81·10-s − 6.35·11-s − 35.1·12-s + 16.4·13-s + 117.·14-s + 8.44·15-s − 79.7·16-s + 8.35·17-s + 120.·18-s − 141.·19-s − 4.85·20-s − 260.·21-s − 22.4·22-s + 123.·23-s + 96.8·24-s − 123.·25-s + 58.2·26-s − 55.5·27-s + 149.·28-s + ⋯ |
| L(s) = 1 | + 1.24·2-s − 1.50·3-s + 0.561·4-s − 0.0965·5-s − 1.88·6-s + 1.79·7-s − 0.547·8-s + 1.26·9-s − 0.120·10-s − 0.174·11-s − 0.845·12-s + 0.351·13-s + 2.24·14-s + 0.145·15-s − 1.24·16-s + 0.119·17-s + 1.57·18-s − 1.70·19-s − 0.0542·20-s − 2.70·21-s − 0.217·22-s + 1.12·23-s + 0.823·24-s − 0.990·25-s + 0.439·26-s − 0.396·27-s + 1.01·28-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 | \( 1 - 3.53T + 8T^{2} \) |
| 3 | \( 1 + 7.81T + 27T^{2} \) |
| 5 | \( 1 + 1.07T + 125T^{2} \) |
| 7 | \( 1 - 33.3T + 343T^{2} \) |
| 11 | \( 1 + 6.35T + 1.33e3T^{2} \) |
| 13 | \( 1 - 16.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 8.35T + 4.91e3T^{2} \) |
| 19 | \( 1 + 141.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 123.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 54.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 253.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 246.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 492.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 335.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 386.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 480.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 264.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 308.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 67.5T + 3.57e5T^{2} \) |
| 73 | \( 1 + 835.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 17.9T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.01e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.24e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 287.T + 9.12e5T^{2} \) |
| 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
−8.367146633496419551923432892326, −7.47231627004140606799433042037, −6.48900716067563408967517567953, −5.76256787483027768165804442608, −5.24086201928732811556213679852, −4.48482120420171511193413731825, −4.05177825254097385805973996727, −2.44906642104685640289605778894, −1.29325988079754641816666644437, 0,
1.29325988079754641816666644437, 2.44906642104685640289605778894, 4.05177825254097385805973996727, 4.48482120420171511193413731825, 5.24086201928732811556213679852, 5.76256787483027768165804442608, 6.48900716067563408967517567953, 7.47231627004140606799433042037, 8.367146633496419551923432892326
