| L(s) = 1 | − 3-s − 3.29·5-s + 3.59·7-s + 9-s + 11-s + 13-s + 3.29·15-s − 8.12·17-s + 3.59·19-s − 3.59·21-s − 3.74·23-s + 5.83·25-s − 27-s + 3.69·29-s + 5.29·31-s − 33-s − 11.8·35-s − 9.82·37-s − 39-s + 8.08·41-s + 2.94·43-s − 3.29·45-s + 8.73·47-s + 5.90·49-s + 8.12·51-s − 8.58·53-s − 3.29·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.47·5-s + 1.35·7-s + 0.333·9-s + 0.301·11-s + 0.277·13-s + 0.849·15-s − 1.97·17-s + 0.824·19-s − 0.784·21-s − 0.781·23-s + 1.16·25-s − 0.192·27-s + 0.686·29-s + 0.950·31-s − 0.174·33-s − 1.99·35-s − 1.61·37-s − 0.160·39-s + 1.26·41-s + 0.448·43-s − 0.490·45-s + 1.27·47-s + 0.844·49-s + 1.13·51-s − 1.17·53-s − 0.443·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1716 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1716 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.126886888\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.126886888\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 3.29T + 5T^{2} \) |
| 7 | \( 1 - 3.59T + 7T^{2} \) |
| 17 | \( 1 + 8.12T + 17T^{2} \) |
| 19 | \( 1 - 3.59T + 19T^{2} \) |
| 23 | \( 1 + 3.74T + 23T^{2} \) |
| 29 | \( 1 - 3.69T + 29T^{2} \) |
| 31 | \( 1 - 5.29T + 31T^{2} \) |
| 37 | \( 1 + 9.82T + 37T^{2} \) |
| 41 | \( 1 - 8.08T + 41T^{2} \) |
| 43 | \( 1 - 2.94T + 43T^{2} \) |
| 47 | \( 1 - 8.73T + 47T^{2} \) |
| 53 | \( 1 + 8.58T + 53T^{2} \) |
| 59 | \( 1 - 6.58T + 59T^{2} \) |
| 61 | \( 1 - 7.24T + 61T^{2} \) |
| 67 | \( 1 + 4.38T + 67T^{2} \) |
| 71 | \( 1 - 9.33T + 71T^{2} \) |
| 73 | \( 1 + 2.99T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 + 4.09T + 83T^{2} \) |
| 89 | \( 1 - 16.5T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−9.094593696769902482194964101903, −8.424316366762449136906964697142, −7.76502398093856125184709578243, −7.06124138343007642308482608929, −6.17395244485580580516967100185, −4.94770118494214736102098544600, −4.45250354126549775719074503917, −3.68366501054217678554885000572, −2.15471131602915029904625325800, −0.76209506166914759635840529390,
0.76209506166914759635840529390, 2.15471131602915029904625325800, 3.68366501054217678554885000572, 4.45250354126549775719074503917, 4.94770118494214736102098544600, 6.17395244485580580516967100185, 7.06124138343007642308482608929, 7.76502398093856125184709578243, 8.424316366762449136906964697142, 9.094593696769902482194964101903
