| L(s) = 1 | + 3.41·3-s + 1.92·5-s − 2.75·7-s + 8.64·9-s + 6.53·13-s + 6.58·15-s − 2.76·17-s − 19-s − 9.41·21-s + 2.27·23-s − 1.27·25-s + 19.2·27-s + 6.08·29-s + 8.85·31-s − 5.32·35-s + 2.35·37-s + 22.3·39-s − 6.33·41-s + 2.14·43-s + 16.6·45-s + 0.191·47-s + 0.615·49-s − 9.44·51-s − 5.00·53-s − 3.41·57-s + 2.73·59-s − 1.56·61-s + ⋯ |
| L(s) = 1 | + 1.97·3-s + 0.862·5-s − 1.04·7-s + 2.88·9-s + 1.81·13-s + 1.69·15-s − 0.671·17-s − 0.229·19-s − 2.05·21-s + 0.475·23-s − 0.255·25-s + 3.70·27-s + 1.12·29-s + 1.59·31-s − 0.899·35-s + 0.386·37-s + 3.57·39-s − 0.989·41-s + 0.326·43-s + 2.48·45-s + 0.0279·47-s + 0.0878·49-s − 1.32·51-s − 0.687·53-s − 0.452·57-s + 0.356·59-s − 0.200·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.730884262\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.730884262\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 3.41T + 3T^{2} \) |
| 5 | \( 1 - 1.92T + 5T^{2} \) |
| 7 | \( 1 + 2.75T + 7T^{2} \) |
| 13 | \( 1 - 6.53T + 13T^{2} \) |
| 17 | \( 1 + 2.76T + 17T^{2} \) |
| 23 | \( 1 - 2.27T + 23T^{2} \) |
| 29 | \( 1 - 6.08T + 29T^{2} \) |
| 31 | \( 1 - 8.85T + 31T^{2} \) |
| 37 | \( 1 - 2.35T + 37T^{2} \) |
| 41 | \( 1 + 6.33T + 41T^{2} \) |
| 43 | \( 1 - 2.14T + 43T^{2} \) |
| 47 | \( 1 - 0.191T + 47T^{2} \) |
| 53 | \( 1 + 5.00T + 53T^{2} \) |
| 59 | \( 1 - 2.73T + 59T^{2} \) |
| 61 | \( 1 + 1.56T + 61T^{2} \) |
| 67 | \( 1 + 14.8T + 67T^{2} \) |
| 71 | \( 1 + 3.69T + 71T^{2} \) |
| 73 | \( 1 + 5.84T + 73T^{2} \) |
| 79 | \( 1 + 7.19T + 79T^{2} \) |
| 83 | \( 1 - 0.835T + 83T^{2} \) |
| 89 | \( 1 + 8.40T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−7.987964619855542126543313062817, −6.98128582299454294460897158526, −6.47953455154100970692392621571, −5.96403056005248800570379102804, −4.61282450950564908497654979192, −4.04356816072538561037639923903, −3.12857355762253281662177707520, −2.86920787370882807440047130134, −1.88204632485773499169785120034, −1.14226683499728202886320947775,
1.14226683499728202886320947775, 1.88204632485773499169785120034, 2.86920787370882807440047130134, 3.12857355762253281662177707520, 4.04356816072538561037639923903, 4.61282450950564908497654979192, 5.96403056005248800570379102804, 6.47953455154100970692392621571, 6.98128582299454294460897158526, 7.987964619855542126543313062817
