| L(s) = 1 | + 2-s + 3-s + 4-s + 3.75·5-s + 6-s − 2.75·7-s + 8-s + 9-s + 3.75·10-s + 1.40·11-s + 12-s − 13-s − 2.75·14-s + 3.75·15-s + 16-s + 4.85·17-s + 18-s − 4.50·19-s + 3.75·20-s − 2.75·21-s + 1.40·22-s − 4.42·23-s + 24-s + 9.06·25-s − 26-s + 27-s − 2.75·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.67·5-s + 0.408·6-s − 1.03·7-s + 0.353·8-s + 0.333·9-s + 1.18·10-s + 0.424·11-s + 0.288·12-s − 0.277·13-s − 0.735·14-s + 0.968·15-s + 0.250·16-s + 1.17·17-s + 0.235·18-s − 1.03·19-s + 0.838·20-s − 0.600·21-s + 0.299·22-s − 0.922·23-s + 0.204·24-s + 1.81·25-s − 0.196·26-s + 0.192·27-s − 0.519·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.540704260\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.540704260\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 5 | \( 1 - 3.75T + 5T^{2} \) |
| 7 | \( 1 + 2.75T + 7T^{2} \) |
| 11 | \( 1 - 1.40T + 11T^{2} \) |
| 17 | \( 1 - 4.85T + 17T^{2} \) |
| 19 | \( 1 + 4.50T + 19T^{2} \) |
| 23 | \( 1 + 4.42T + 23T^{2} \) |
| 29 | \( 1 - 6.19T + 29T^{2} \) |
| 31 | \( 1 + 8.77T + 31T^{2} \) |
| 37 | \( 1 - 4.12T + 37T^{2} \) |
| 41 | \( 1 + 1.99T + 41T^{2} \) |
| 43 | \( 1 - 11.5T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 - 6.40T + 53T^{2} \) |
| 59 | \( 1 + 9.19T + 59T^{2} \) |
| 61 | \( 1 - 14.5T + 61T^{2} \) |
| 67 | \( 1 - 6.93T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 + 5.76T + 73T^{2} \) |
| 79 | \( 1 + 3.87T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 7.07T + 89T^{2} \) |
| 97 | \( 1 - 17.8T + 97T^{2} \) |
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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
−7.69536308337205791457403845679, −6.89903381940937734610102665256, −6.30351297344089742257825016052, −5.82633621476377193700473415594, −5.20085606501376773512991160037, −4.12989325308956360010264027606, −3.52082225899014846537053414338, −2.52367573401725771736998626073, −2.19011811165107830292923573546, −1.05262784849918099496942219699,
1.05262784849918099496942219699, 2.19011811165107830292923573546, 2.52367573401725771736998626073, 3.52082225899014846537053414338, 4.12989325308956360010264027606, 5.20085606501376773512991160037, 5.82633621476377193700473415594, 6.30351297344089742257825016052, 6.89903381940937734610102665256, 7.69536308337205791457403845679
