| L(s) = 1 | − 1.35·3-s + 0.648·7-s − 1.17·9-s + 3.35·11-s − 4.17·13-s − 4.82·17-s + 6.82·19-s − 0.876·21-s − 5.52·23-s + 5.64·27-s − 29-s − 2.82·31-s − 4.53·33-s + 10.2·37-s + 5.64·39-s + 8.17·41-s + 5.69·43-s + 2.64·47-s − 6.58·49-s + 6.51·51-s + 2.87·53-s − 9.22·57-s − 13.2·59-s − 1.12·61-s − 0.759·63-s − 1.52·67-s + 7.46·69-s + ⋯ |
| L(s) = 1 | − 0.780·3-s + 0.244·7-s − 0.390·9-s + 1.01·11-s − 1.15·13-s − 1.16·17-s + 1.56·19-s − 0.191·21-s − 1.15·23-s + 1.08·27-s − 0.185·29-s − 0.506·31-s − 0.788·33-s + 1.68·37-s + 0.903·39-s + 1.27·41-s + 0.868·43-s + 0.386·47-s − 0.940·49-s + 0.912·51-s + 0.395·53-s − 1.22·57-s − 1.72·59-s − 0.143·61-s − 0.0957·63-s − 0.186·67-s + 0.899·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.143378068\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.143378068\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 1.35T + 3T^{2} \) |
| 7 | \( 1 - 0.648T + 7T^{2} \) |
| 11 | \( 1 - 3.35T + 11T^{2} \) |
| 13 | \( 1 + 4.17T + 13T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 19 | \( 1 - 6.82T + 19T^{2} \) |
| 23 | \( 1 + 5.52T + 23T^{2} \) |
| 31 | \( 1 + 2.82T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 - 8.17T + 41T^{2} \) |
| 43 | \( 1 - 5.69T + 43T^{2} \) |
| 47 | \( 1 - 2.64T + 47T^{2} \) |
| 53 | \( 1 - 2.87T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 + 1.12T + 61T^{2} \) |
| 67 | \( 1 + 1.52T + 67T^{2} \) |
| 71 | \( 1 + 8.87T + 71T^{2} \) |
| 73 | \( 1 + 9.69T + 73T^{2} \) |
| 79 | \( 1 - 8.99T + 79T^{2} \) |
| 83 | \( 1 + 1.94T + 83T^{2} \) |
| 89 | \( 1 - 17.0T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
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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
−9.015977192396361319872840442850, −7.78642298182288257474753851501, −7.33359183685454525661575472491, −6.25596899791561431184802044322, −5.86638111013525184149165881527, −4.84183093577182791775690800991, −4.27851545059666153150086605784, −3.06718601321511281743720444891, −2.02726486379836427945556783237, −0.67424163363638484228142578845,
0.67424163363638484228142578845, 2.02726486379836427945556783237, 3.06718601321511281743720444891, 4.27851545059666153150086605784, 4.84183093577182791775690800991, 5.86638111013525184149165881527, 6.25596899791561431184802044322, 7.33359183685454525661575472491, 7.78642298182288257474753851501, 9.015977192396361319872840442850
