| L(s) = 1 | + 1.29·2-s − 1.73·3-s − 0.311·4-s − 5-s − 2.25·6-s − 3.87·7-s − 3.00·8-s + 0.0154·9-s − 1.29·10-s − 1.17·11-s + 0.540·12-s − 5.85·13-s − 5.03·14-s + 1.73·15-s − 3.28·16-s + 5.19·17-s + 0.0200·18-s + 4.16·19-s + 0.311·20-s + 6.72·21-s − 1.53·22-s − 8.14·23-s + 5.21·24-s + 25-s − 7.60·26-s + 5.18·27-s + 1.20·28-s + ⋯ |
| L(s) = 1 | + 0.918·2-s − 1.00·3-s − 0.155·4-s − 0.447·5-s − 0.921·6-s − 1.46·7-s − 1.06·8-s + 0.00515·9-s − 0.410·10-s − 0.355·11-s + 0.155·12-s − 1.62·13-s − 1.34·14-s + 0.448·15-s − 0.820·16-s + 1.25·17-s + 0.00473·18-s + 0.954·19-s + 0.0695·20-s + 1.46·21-s − 0.326·22-s − 1.69·23-s + 1.06·24-s + 0.200·25-s − 1.49·26-s + 0.997·27-s + 0.227·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4504590480\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4504590480\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 1.29T + 2T^{2} \) |
| 3 | \( 1 + 1.73T + 3T^{2} \) |
| 7 | \( 1 + 3.87T + 7T^{2} \) |
| 11 | \( 1 + 1.17T + 11T^{2} \) |
| 13 | \( 1 + 5.85T + 13T^{2} \) |
| 17 | \( 1 - 5.19T + 17T^{2} \) |
| 19 | \( 1 - 4.16T + 19T^{2} \) |
| 23 | \( 1 + 8.14T + 23T^{2} \) |
| 29 | \( 1 + 1.03T + 29T^{2} \) |
| 31 | \( 1 - 2.43T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 - 2.14T + 41T^{2} \) |
| 43 | \( 1 - 5.27T + 43T^{2} \) |
| 47 | \( 1 - 2.08T + 47T^{2} \) |
| 53 | \( 1 - 0.828T + 53T^{2} \) |
| 59 | \( 1 + 0.437T + 59T^{2} \) |
| 61 | \( 1 - 4.17T + 61T^{2} \) |
| 67 | \( 1 - 1.00T + 67T^{2} \) |
| 71 | \( 1 + 0.747T + 71T^{2} \) |
| 73 | \( 1 + 1.42T + 73T^{2} \) |
| 79 | \( 1 - 1.34T + 79T^{2} \) |
| 83 | \( 1 - 0.0187T + 83T^{2} \) |
| 89 | \( 1 + 3.97T + 89T^{2} \) |
| 97 | \( 1 + 7.25T + 97T^{2} \) |
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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
−9.502516251808513842163740632353, −8.275658168727192022716710862424, −7.32174983724907905376466715720, −6.56945004091042957669829069880, −5.58126528223276803491246187810, −5.41898890155309384403900809531, −4.32662342713186409574937271009, −3.41398385607141144732493993112, −2.70590618659600682301183089415, −0.38231412699379024657865473821,
0.38231412699379024657865473821, 2.70590618659600682301183089415, 3.41398385607141144732493993112, 4.32662342713186409574937271009, 5.41898890155309384403900809531, 5.58126528223276803491246187810, 6.56945004091042957669829069880, 7.32174983724907905376466715720, 8.275658168727192022716710862424, 9.502516251808513842163740632353
