| L(s) = 1 | − 2.08·2-s + 3-s + 2.35·4-s − 2.57·5-s − 2.08·6-s − 7-s − 0.751·8-s + 9-s + 5.38·10-s + 2.22·11-s + 2.35·12-s + 3.69·13-s + 2.08·14-s − 2.57·15-s − 3.15·16-s + 7.02·17-s − 2.08·18-s − 7.75·19-s − 6.08·20-s − 21-s − 4.64·22-s + 5.23·23-s − 0.751·24-s + 1.64·25-s − 7.71·26-s + 27-s − 2.35·28-s + ⋯ |
| L(s) = 1 | − 1.47·2-s + 0.577·3-s + 1.17·4-s − 1.15·5-s − 0.852·6-s − 0.377·7-s − 0.265·8-s + 0.333·9-s + 1.70·10-s + 0.670·11-s + 0.681·12-s + 1.02·13-s + 0.558·14-s − 0.665·15-s − 0.787·16-s + 1.70·17-s − 0.492·18-s − 1.78·19-s − 1.36·20-s − 0.218·21-s − 0.989·22-s + 1.09·23-s − 0.153·24-s + 0.329·25-s − 1.51·26-s + 0.192·27-s − 0.445·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8523873020\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8523873020\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 + T \) |
| good | 2 | \( 1 + 2.08T + 2T^{2} \) |
| 5 | \( 1 + 2.57T + 5T^{2} \) |
| 11 | \( 1 - 2.22T + 11T^{2} \) |
| 13 | \( 1 - 3.69T + 13T^{2} \) |
| 17 | \( 1 - 7.02T + 17T^{2} \) |
| 19 | \( 1 + 7.75T + 19T^{2} \) |
| 23 | \( 1 - 5.23T + 23T^{2} \) |
| 29 | \( 1 + 7.88T + 29T^{2} \) |
| 31 | \( 1 - 5.46T + 31T^{2} \) |
| 37 | \( 1 - 3.57T + 37T^{2} \) |
| 41 | \( 1 + 9.26T + 41T^{2} \) |
| 43 | \( 1 - 6.03T + 43T^{2} \) |
| 47 | \( 1 - 12.2T + 47T^{2} \) |
| 53 | \( 1 + 2.08T + 53T^{2} \) |
| 59 | \( 1 - 9.76T + 59T^{2} \) |
| 61 | \( 1 + 7.73T + 61T^{2} \) |
| 67 | \( 1 - 1.90T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 5.81T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 + 7.41T + 83T^{2} \) |
| 89 | \( 1 + 8.25T + 89T^{2} \) |
| 97 | \( 1 - 0.695T + 97T^{2} \) |
| 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
−8.458404258485405818645935723850, −7.981507170180132147646458256077, −7.27331325789435419571695114985, −6.68778944892049586707106959105, −5.72116101364249007280799932013, −4.27890263871754852830311331603, −3.80015612822848066824134970568, −2.85070718583378941692624860491, −1.59874203595099135603440721948, −0.67927484478353068322475908228,
0.67927484478353068322475908228, 1.59874203595099135603440721948, 2.85070718583378941692624860491, 3.80015612822848066824134970568, 4.27890263871754852830311331603, 5.72116101364249007280799932013, 6.68778944892049586707106959105, 7.27331325789435419571695114985, 7.981507170180132147646458256077, 8.458404258485405818645935723850
