L(s) = 1 | + 3-s + 5-s − 4.83·7-s + 9-s + 2.25·11-s − 13-s + 15-s + 6.32·17-s − 2.58·19-s − 4.83·21-s + 4.83·23-s + 25-s + 27-s + 9.09·29-s − 0.510·31-s + 2.25·33-s − 4.83·35-s + 4.25·37-s − 39-s + 0.255·41-s + 9.16·43-s + 45-s − 4.51·47-s + 16.4·49-s + 6.32·51-s − 9.93·53-s + 2.25·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.82·7-s + 0.333·9-s + 0.679·11-s − 0.277·13-s + 0.258·15-s + 1.53·17-s − 0.592·19-s − 1.05·21-s + 1.00·23-s + 0.200·25-s + 0.192·27-s + 1.68·29-s − 0.0916·31-s + 0.392·33-s − 0.818·35-s + 0.699·37-s − 0.160·39-s + 0.0398·41-s + 1.39·43-s + 0.149·45-s − 0.657·47-s + 2.34·49-s + 0.886·51-s − 1.36·53-s + 0.304·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.003641031\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.003641031\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + 4.83T + 7T^{2} \) |
| 11 | \( 1 - 2.25T + 11T^{2} \) |
| 17 | \( 1 - 6.32T + 17T^{2} \) |
| 19 | \( 1 + 2.58T + 19T^{2} \) |
| 23 | \( 1 - 4.83T + 23T^{2} \) |
| 29 | \( 1 - 9.09T + 29T^{2} \) |
| 31 | \( 1 + 0.510T + 31T^{2} \) |
| 37 | \( 1 - 4.25T + 37T^{2} \) |
| 41 | \( 1 - 0.255T + 41T^{2} \) |
| 43 | \( 1 - 9.16T + 43T^{2} \) |
| 47 | \( 1 + 4.51T + 47T^{2} \) |
| 53 | \( 1 + 9.93T + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 - 9.93T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 + 5.74T + 71T^{2} \) |
| 73 | \( 1 - 2.90T + 73T^{2} \) |
| 79 | \( 1 - 5.74T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 - 3.74T + 89T^{2} \) |
| 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
−9.501113302480848460264710932406, −8.858408002199664429084022054431, −7.86087682310024991489931890332, −6.84138808971810097680348953488, −6.39341778376821908333162184638, −5.44628283922666503759763070940, −4.18267650448002378648705745008, −3.24146399340038109614269410199, −2.62453354036007584172383879859, −1.00391181317431803394670358994,
1.00391181317431803394670358994, 2.62453354036007584172383879859, 3.24146399340038109614269410199, 4.18267650448002378648705745008, 5.44628283922666503759763070940, 6.39341778376821908333162184638, 6.84138808971810097680348953488, 7.86087682310024991489931890332, 8.858408002199664429084022054431, 9.501113302480848460264710932406