L(s) = 1 | − 2.04·2-s + 3-s + 2.16·4-s + 3.68·5-s − 2.04·6-s − 0.345·8-s + 9-s − 7.53·10-s + 0.232·11-s + 2.16·12-s − 2.59·13-s + 3.68·15-s − 3.63·16-s − 4.71·17-s − 2.04·18-s + 7.74·19-s + 8.00·20-s − 0.474·22-s + 8.48·23-s − 0.345·24-s + 8.61·25-s + 5.29·26-s + 27-s − 4.10·29-s − 7.53·30-s + 10.4·31-s + 8.10·32-s + ⋯ |
L(s) = 1 | − 1.44·2-s + 0.577·3-s + 1.08·4-s + 1.64·5-s − 0.833·6-s − 0.122·8-s + 0.333·9-s − 2.38·10-s + 0.0700·11-s + 0.626·12-s − 0.718·13-s + 0.952·15-s − 0.908·16-s − 1.14·17-s − 0.481·18-s + 1.77·19-s + 1.78·20-s − 0.101·22-s + 1.76·23-s − 0.0704·24-s + 1.72·25-s + 1.03·26-s + 0.192·27-s − 0.763·29-s − 1.37·30-s + 1.87·31-s + 1.43·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.766286634\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.766286634\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 2 | \( 1 + 2.04T + 2T^{2} \) |
| 5 | \( 1 - 3.68T + 5T^{2} \) |
| 11 | \( 1 - 0.232T + 11T^{2} \) |
| 13 | \( 1 + 2.59T + 13T^{2} \) |
| 17 | \( 1 + 4.71T + 17T^{2} \) |
| 19 | \( 1 - 7.74T + 19T^{2} \) |
| 23 | \( 1 - 8.48T + 23T^{2} \) |
| 29 | \( 1 + 4.10T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 + 8.13T + 37T^{2} \) |
| 43 | \( 1 - 2.59T + 43T^{2} \) |
| 47 | \( 1 + 3.41T + 47T^{2} \) |
| 53 | \( 1 + 3.35T + 53T^{2} \) |
| 59 | \( 1 - 8.02T + 59T^{2} \) |
| 61 | \( 1 + 13.1T + 61T^{2} \) |
| 67 | \( 1 - 6.60T + 67T^{2} \) |
| 71 | \( 1 - 3.23T + 71T^{2} \) |
| 73 | \( 1 - 6.63T + 73T^{2} \) |
| 79 | \( 1 + 5.24T + 79T^{2} \) |
| 83 | \( 1 + 2.50T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 - 7.90T + 97T^{2} \) |
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
−8.346895970523962091559250602823, −7.35504598312377920342421008307, −6.98926072128907067022330537448, −6.23248529515282588351325027455, −5.19022016590142263323383843832, −4.68562872078970228764487777701, −3.15065839005923606214549796080, −2.45526706181324065171226750647, −1.70854597789136532611494387765, −0.891506266224048905905734494936,
0.891506266224048905905734494936, 1.70854597789136532611494387765, 2.45526706181324065171226750647, 3.15065839005923606214549796080, 4.68562872078970228764487777701, 5.19022016590142263323383843832, 6.23248529515282588351325027455, 6.98926072128907067022330537448, 7.35504598312377920342421008307, 8.346895970523962091559250602823