| L(s) = 1 | − 2-s + 3-s + 4-s + 0.374·5-s − 6-s − 3.42·7-s − 8-s + 9-s − 0.374·10-s − 3.51·11-s + 12-s − 2.41·13-s + 3.42·14-s + 0.374·15-s + 16-s + 17-s − 18-s + 3.61·19-s + 0.374·20-s − 3.42·21-s + 3.51·22-s − 0.841·23-s − 24-s − 4.85·25-s + 2.41·26-s + 27-s − 3.42·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.167·5-s − 0.408·6-s − 1.29·7-s − 0.353·8-s + 0.333·9-s − 0.118·10-s − 1.06·11-s + 0.288·12-s − 0.669·13-s + 0.914·14-s + 0.0966·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s + 0.830·19-s + 0.0836·20-s − 0.746·21-s + 0.750·22-s − 0.175·23-s − 0.204·24-s − 0.971·25-s + 0.473·26-s + 0.192·27-s − 0.646·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.012831570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.012831570\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 5 | \( 1 - 0.374T + 5T^{2} \) |
| 7 | \( 1 + 3.42T + 7T^{2} \) |
| 11 | \( 1 + 3.51T + 11T^{2} \) |
| 13 | \( 1 + 2.41T + 13T^{2} \) |
| 19 | \( 1 - 3.61T + 19T^{2} \) |
| 23 | \( 1 + 0.841T + 23T^{2} \) |
| 29 | \( 1 + 1.69T + 29T^{2} \) |
| 31 | \( 1 + 8.51T + 31T^{2} \) |
| 37 | \( 1 - 8.12T + 37T^{2} \) |
| 41 | \( 1 - 0.162T + 41T^{2} \) |
| 43 | \( 1 - 7.74T + 43T^{2} \) |
| 47 | \( 1 - 7.55T + 47T^{2} \) |
| 53 | \( 1 + 1.45T + 53T^{2} \) |
| 61 | \( 1 + 0.959T + 61T^{2} \) |
| 67 | \( 1 - 2.70T + 67T^{2} \) |
| 71 | \( 1 + 3.20T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 + 4.68T + 83T^{2} \) |
| 89 | \( 1 - 0.656T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−7.906650631932137998334315254546, −7.54137370956703562850537229442, −6.95174775726921665028135055539, −5.91539447472229890142159732548, −5.51067450433127122273151976715, −4.27356648712871979164829066375, −3.34601612331522183935925845057, −2.72472786851801657652489953587, −1.97268285896971531200014845711, −0.54079908865446791115528818691,
0.54079908865446791115528818691, 1.97268285896971531200014845711, 2.72472786851801657652489953587, 3.34601612331522183935925845057, 4.27356648712871979164829066375, 5.51067450433127122273151976715, 5.91539447472229890142159732548, 6.95174775726921665028135055539, 7.54137370956703562850537229442, 7.906650631932137998334315254546
