| L(s) = 1 | − 2-s + 4-s + 0.473·5-s − 8-s − 0.473·10-s − 11-s + 7.03·13-s + 16-s + 4.64·17-s + 6.55·19-s + 0.473·20-s + 22-s − 1.49·23-s − 4.77·25-s − 7.03·26-s + 2.11·29-s + 6.83·31-s − 32-s − 4.64·34-s − 4.32·37-s − 6.55·38-s − 0.473·40-s + 11.2·41-s + 0.158·43-s − 44-s + 1.49·46-s − 0.270·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.211·5-s − 0.353·8-s − 0.149·10-s − 0.301·11-s + 1.95·13-s + 0.250·16-s + 1.12·17-s + 1.50·19-s + 0.105·20-s + 0.213·22-s − 0.312·23-s − 0.955·25-s − 1.37·26-s + 0.393·29-s + 1.22·31-s − 0.176·32-s − 0.796·34-s − 0.711·37-s − 1.06·38-s − 0.0748·40-s + 1.75·41-s + 0.0241·43-s − 0.150·44-s + 0.220·46-s − 0.0394·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.020370371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.020370371\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 0.473T + 5T^{2} \) |
| 13 | \( 1 - 7.03T + 13T^{2} \) |
| 17 | \( 1 - 4.64T + 17T^{2} \) |
| 19 | \( 1 - 6.55T + 19T^{2} \) |
| 23 | \( 1 + 1.49T + 23T^{2} \) |
| 29 | \( 1 - 2.11T + 29T^{2} \) |
| 31 | \( 1 - 6.83T + 31T^{2} \) |
| 37 | \( 1 + 4.32T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 - 0.158T + 43T^{2} \) |
| 47 | \( 1 + 0.270T + 47T^{2} \) |
| 53 | \( 1 - 9.66T + 53T^{2} \) |
| 59 | \( 1 + 9.82T + 59T^{2} \) |
| 61 | \( 1 - 15.1T + 61T^{2} \) |
| 67 | \( 1 + 1.21T + 67T^{2} \) |
| 71 | \( 1 + 15.0T + 71T^{2} \) |
| 73 | \( 1 + 6.13T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 - 7.54T + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−7.73146662358026580679163714062, −7.22135846105435075861225490138, −6.12823078120035590030121501004, −5.93250233921562734951934839974, −5.11771542428304673747607150832, −3.97705405554895215965815780523, −3.36553942625143503710044523424, −2.55373992306927247605057385042, −1.40925759376384295005214542228, −0.856578081408337407676950322815,
0.856578081408337407676950322815, 1.40925759376384295005214542228, 2.55373992306927247605057385042, 3.36553942625143503710044523424, 3.97705405554895215965815780523, 5.11771542428304673747607150832, 5.93250233921562734951934839974, 6.12823078120035590030121501004, 7.22135846105435075861225490138, 7.73146662358026580679163714062
