| L(s) = 1 | − 2-s + 1.96·3-s + 4-s − 1.72·5-s − 1.96·6-s − 4.09·7-s − 8-s + 0.851·9-s + 1.72·10-s + 0.354·11-s + 1.96·12-s + 5.40·13-s + 4.09·14-s − 3.39·15-s + 16-s − 4.29·17-s − 0.851·18-s + 3.40·19-s − 1.72·20-s − 8.04·21-s − 0.354·22-s − 5.29·23-s − 1.96·24-s − 2.01·25-s − 5.40·26-s − 4.21·27-s − 4.09·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.13·3-s + 0.5·4-s − 0.772·5-s − 0.801·6-s − 1.54·7-s − 0.353·8-s + 0.283·9-s + 0.546·10-s + 0.106·11-s + 0.566·12-s + 1.50·13-s + 1.09·14-s − 0.875·15-s + 0.250·16-s − 1.04·17-s − 0.200·18-s + 0.780·19-s − 0.386·20-s − 1.75·21-s − 0.0756·22-s − 1.10·23-s − 0.400·24-s − 0.403·25-s − 1.06·26-s − 0.811·27-s − 0.774·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.196681668\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.196681668\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2011 | \( 1 - T \) |
| good | 3 | \( 1 - 1.96T + 3T^{2} \) |
| 5 | \( 1 + 1.72T + 5T^{2} \) |
| 7 | \( 1 + 4.09T + 7T^{2} \) |
| 11 | \( 1 - 0.354T + 11T^{2} \) |
| 13 | \( 1 - 5.40T + 13T^{2} \) |
| 17 | \( 1 + 4.29T + 17T^{2} \) |
| 19 | \( 1 - 3.40T + 19T^{2} \) |
| 23 | \( 1 + 5.29T + 23T^{2} \) |
| 29 | \( 1 - 7.63T + 29T^{2} \) |
| 31 | \( 1 + 8.76T + 31T^{2} \) |
| 37 | \( 1 - 8.75T + 37T^{2} \) |
| 41 | \( 1 - 6.02T + 41T^{2} \) |
| 43 | \( 1 - 6.86T + 43T^{2} \) |
| 47 | \( 1 + 4.01T + 47T^{2} \) |
| 53 | \( 1 - 3.00T + 53T^{2} \) |
| 59 | \( 1 + 6.89T + 59T^{2} \) |
| 61 | \( 1 - 7.62T + 61T^{2} \) |
| 67 | \( 1 + 3.55T + 67T^{2} \) |
| 71 | \( 1 + 3.91T + 71T^{2} \) |
| 73 | \( 1 - 5.20T + 73T^{2} \) |
| 79 | \( 1 - 2.21T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 - 2.42T + 89T^{2} \) |
| 97 | \( 1 - 0.628T + 97T^{2} \) |
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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
−8.483751898141256444684229187092, −7.88370800868326483013136421646, −7.24385779399012852536330602359, −6.30901831365349711575247982849, −5.90313769148181308763974504390, −4.19384194966346252638074218113, −3.61964140717020439006312131754, −3.00553692427186299982046399105, −2.09629089129330400747003733451, −0.63328828425465858016314893157,
0.63328828425465858016314893157, 2.09629089129330400747003733451, 3.00553692427186299982046399105, 3.61964140717020439006312131754, 4.19384194966346252638074218113, 5.90313769148181308763974504390, 6.30901831365349711575247982849, 7.24385779399012852536330602359, 7.88370800868326483013136421646, 8.483751898141256444684229187092
