| L(s) = 1 | − 1.10·2-s − 3-s − 0.784·4-s + 0.760·5-s + 1.10·6-s + 7-s + 3.06·8-s + 9-s − 0.838·10-s + 0.597·11-s + 0.784·12-s − 4.75·13-s − 1.10·14-s − 0.760·15-s − 1.81·16-s + 1.90·17-s − 1.10·18-s − 5.87·19-s − 0.596·20-s − 21-s − 0.658·22-s − 7.97·23-s − 3.06·24-s − 4.42·25-s + 5.24·26-s − 27-s − 0.784·28-s + ⋯ |
| L(s) = 1 | − 0.779·2-s − 0.577·3-s − 0.392·4-s + 0.340·5-s + 0.450·6-s + 0.377·7-s + 1.08·8-s + 0.333·9-s − 0.265·10-s + 0.180·11-s + 0.226·12-s − 1.31·13-s − 0.294·14-s − 0.196·15-s − 0.454·16-s + 0.462·17-s − 0.259·18-s − 1.34·19-s − 0.133·20-s − 0.218·21-s − 0.140·22-s − 1.66·23-s − 0.626·24-s − 0.884·25-s + 1.02·26-s − 0.192·27-s − 0.148·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4759821630\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4759821630\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 + 1.10T + 2T^{2} \) |
| 5 | \( 1 - 0.760T + 5T^{2} \) |
| 11 | \( 1 - 0.597T + 11T^{2} \) |
| 13 | \( 1 + 4.75T + 13T^{2} \) |
| 17 | \( 1 - 1.90T + 17T^{2} \) |
| 19 | \( 1 + 5.87T + 19T^{2} \) |
| 23 | \( 1 + 7.97T + 23T^{2} \) |
| 29 | \( 1 - 3.85T + 29T^{2} \) |
| 31 | \( 1 + 4.54T + 31T^{2} \) |
| 37 | \( 1 + 4.13T + 37T^{2} \) |
| 41 | \( 1 - 9.73T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 + 5.17T + 47T^{2} \) |
| 53 | \( 1 - 14.0T + 53T^{2} \) |
| 59 | \( 1 + 7.09T + 59T^{2} \) |
| 61 | \( 1 + 4.13T + 61T^{2} \) |
| 67 | \( 1 - 1.80T + 67T^{2} \) |
| 71 | \( 1 + 15.7T + 71T^{2} \) |
| 73 | \( 1 - 6.46T + 73T^{2} \) |
| 79 | \( 1 + 2.06T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 - 5.04T + 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
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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
−7.79250236733534017264076251506, −7.40085396913881046367519446467, −6.44760214407993002075851701255, −5.79134690277736270855685245233, −4.98561411797075498134352492014, −4.44140039963535464897443610324, −3.69541183082273199276987635139, −2.22834009277939260312354179018, −1.69274299918259500181182963564, −0.39545914663096894956811092649,
0.39545914663096894956811092649, 1.69274299918259500181182963564, 2.22834009277939260312354179018, 3.69541183082273199276987635139, 4.44140039963535464897443610324, 4.98561411797075498134352492014, 5.79134690277736270855685245233, 6.44760214407993002075851701255, 7.40085396913881046367519446467, 7.79250236733534017264076251506
