| L(s) = 1 | + 2.67·2-s − 1.58·3-s + 5.17·4-s + 0.0973·5-s − 4.23·6-s + 0.0377·7-s + 8.51·8-s − 0.498·9-s + 0.260·10-s + 2.07·11-s − 8.18·12-s + 2.37·13-s + 0.101·14-s − 0.153·15-s + 12.4·16-s + 4.23·17-s − 1.33·18-s + 5.29·19-s + 0.503·20-s − 0.0596·21-s + 5.55·22-s − 6.91·23-s − 13.4·24-s − 4.99·25-s + 6.35·26-s + 5.53·27-s + 0.195·28-s + ⋯ |
| L(s) = 1 | + 1.89·2-s − 0.913·3-s + 2.58·4-s + 0.0435·5-s − 1.72·6-s + 0.0142·7-s + 3.00·8-s − 0.166·9-s + 0.0824·10-s + 0.625·11-s − 2.36·12-s + 0.657·13-s + 0.0269·14-s − 0.0397·15-s + 3.11·16-s + 1.02·17-s − 0.314·18-s + 1.21·19-s + 0.112·20-s − 0.0130·21-s + 1.18·22-s − 1.44·23-s − 2.74·24-s − 0.998·25-s + 1.24·26-s + 1.06·27-s + 0.0368·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.901251169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.901251169\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 \) |
| good | 2 | \( 1 - 2.67T + 2T^{2} \) |
| 3 | \( 1 + 1.58T + 3T^{2} \) |
| 5 | \( 1 - 0.0973T + 5T^{2} \) |
| 7 | \( 1 - 0.0377T + 7T^{2} \) |
| 11 | \( 1 - 2.07T + 11T^{2} \) |
| 13 | \( 1 - 2.37T + 13T^{2} \) |
| 17 | \( 1 - 4.23T + 17T^{2} \) |
| 19 | \( 1 - 5.29T + 19T^{2} \) |
| 23 | \( 1 + 6.91T + 23T^{2} \) |
| 31 | \( 1 - 1.44T + 31T^{2} \) |
| 37 | \( 1 + 8.75T + 37T^{2} \) |
| 41 | \( 1 - 3.66T + 41T^{2} \) |
| 43 | \( 1 - 2.56T + 43T^{2} \) |
| 47 | \( 1 + 7.21T + 47T^{2} \) |
| 53 | \( 1 - 4.32T + 53T^{2} \) |
| 59 | \( 1 + 5.71T + 59T^{2} \) |
| 61 | \( 1 + 6.58T + 61T^{2} \) |
| 67 | \( 1 + 1.49T + 67T^{2} \) |
| 71 | \( 1 - 5.92T + 71T^{2} \) |
| 73 | \( 1 + 5.50T + 73T^{2} \) |
| 79 | \( 1 - 2.04T + 79T^{2} \) |
| 83 | \( 1 + 9.54T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 - 0.263T + 97T^{2} \) |
| 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
−10.64118414300011785439852452904, −9.734263808933849848267239907876, −8.143593191629855246209413668962, −7.17934286264439568297156589767, −6.15423043635102841668551995989, −5.80133507376136414707557989706, −4.97967894570630313309309558780, −3.91852354897452376863342461643, −3.12254658908643189036001061818, −1.56751345940846327031364100495,
1.56751345940846327031364100495, 3.12254658908643189036001061818, 3.91852354897452376863342461643, 4.97967894570630313309309558780, 5.80133507376136414707557989706, 6.15423043635102841668551995989, 7.17934286264439568297156589767, 8.143593191629855246209413668962, 9.734263808933849848267239907876, 10.64118414300011785439852452904
