L(s) = 1 | − 2-s − 3.00·3-s + 4-s + 2.50·5-s + 3.00·6-s − 2.50·7-s − 8-s + 6.03·9-s − 2.50·10-s − 1.29·11-s − 3.00·12-s − 6.66·13-s + 2.50·14-s − 7.53·15-s + 16-s + 1.24·17-s − 6.03·18-s + 8.26·19-s + 2.50·20-s + 7.53·21-s + 1.29·22-s − 23-s + 3.00·24-s + 1.28·25-s + 6.66·26-s − 9.13·27-s − 2.50·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.73·3-s + 0.5·4-s + 1.12·5-s + 1.22·6-s − 0.947·7-s − 0.353·8-s + 2.01·9-s − 0.792·10-s − 0.390·11-s − 0.867·12-s − 1.84·13-s + 0.670·14-s − 1.94·15-s + 0.250·16-s + 0.301·17-s − 1.42·18-s + 1.89·19-s + 0.560·20-s + 1.64·21-s + 0.275·22-s − 0.208·23-s + 0.613·24-s + 0.256·25-s + 1.30·26-s − 1.75·27-s − 0.473·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4085681680\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4085681680\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
good | 3 | \( 1 + 3.00T + 3T^{2} \) |
| 5 | \( 1 - 2.50T + 5T^{2} \) |
| 7 | \( 1 + 2.50T + 7T^{2} \) |
| 11 | \( 1 + 1.29T + 11T^{2} \) |
| 13 | \( 1 + 6.66T + 13T^{2} \) |
| 17 | \( 1 - 1.24T + 17T^{2} \) |
| 19 | \( 1 - 8.26T + 19T^{2} \) |
| 29 | \( 1 + 9.40T + 29T^{2} \) |
| 31 | \( 1 - 0.137T + 31T^{2} \) |
| 37 | \( 1 + 8.52T + 37T^{2} \) |
| 41 | \( 1 + 4.48T + 41T^{2} \) |
| 43 | \( 1 - 2.37T + 43T^{2} \) |
| 47 | \( 1 - 9.29T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 - 13.0T + 61T^{2} \) |
| 67 | \( 1 - 3.80T + 67T^{2} \) |
| 71 | \( 1 + 6.69T + 71T^{2} \) |
| 73 | \( 1 - 4.66T + 73T^{2} \) |
| 79 | \( 1 + 8.73T + 79T^{2} \) |
| 83 | \( 1 + 0.628T + 83T^{2} \) |
| 89 | \( 1 - 9.00T + 89T^{2} \) |
| 97 | \( 1 + 12.5T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.78115093815426032270473425194, −7.12234103074219292613396431265, −6.78666169928153212734541863952, −5.74526132408614854192279536655, −5.51501634252172531582581315064, −4.93975774293406260466264987675, −3.57336390871462009092923538198, −2.50313108665885133979809314109, −1.57458870463132293820838519353, −0.40882945504593744074857449718,
0.40882945504593744074857449718, 1.57458870463132293820838519353, 2.50313108665885133979809314109, 3.57336390871462009092923538198, 4.93975774293406260466264987675, 5.51501634252172531582581315064, 5.74526132408614854192279536655, 6.78666169928153212734541863952, 7.12234103074219292613396431265, 7.78115093815426032270473425194