L(s) = 1 | + 2-s + 1.99·3-s + 4-s + 3.41·5-s + 1.99·6-s + 4.21·7-s + 8-s + 0.971·9-s + 3.41·10-s − 3.81·11-s + 1.99·12-s + 2.36·13-s + 4.21·14-s + 6.79·15-s + 16-s + 4.97·17-s + 0.971·18-s − 6.46·19-s + 3.41·20-s + 8.40·21-s − 3.81·22-s − 2.22·23-s + 1.99·24-s + 6.63·25-s + 2.36·26-s − 4.04·27-s + 4.21·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.15·3-s + 0.5·4-s + 1.52·5-s + 0.813·6-s + 1.59·7-s + 0.353·8-s + 0.323·9-s + 1.07·10-s − 1.14·11-s + 0.575·12-s + 0.655·13-s + 1.12·14-s + 1.75·15-s + 0.250·16-s + 1.20·17-s + 0.229·18-s − 1.48·19-s + 0.762·20-s + 1.83·21-s − 0.812·22-s − 0.463·23-s + 0.406·24-s + 1.32·25-s + 0.463·26-s − 0.777·27-s + 0.796·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.765924202\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.765924202\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 2003 | \( 1 + T \) |
good | 3 | \( 1 - 1.99T + 3T^{2} \) |
| 5 | \( 1 - 3.41T + 5T^{2} \) |
| 7 | \( 1 - 4.21T + 7T^{2} \) |
| 11 | \( 1 + 3.81T + 11T^{2} \) |
| 13 | \( 1 - 2.36T + 13T^{2} \) |
| 17 | \( 1 - 4.97T + 17T^{2} \) |
| 19 | \( 1 + 6.46T + 19T^{2} \) |
| 23 | \( 1 + 2.22T + 23T^{2} \) |
| 29 | \( 1 + 6.98T + 29T^{2} \) |
| 31 | \( 1 - 4.75T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 + 1.15T + 41T^{2} \) |
| 43 | \( 1 + 0.504T + 43T^{2} \) |
| 47 | \( 1 - 2.23T + 47T^{2} \) |
| 53 | \( 1 - 0.366T + 53T^{2} \) |
| 59 | \( 1 - 6.24T + 59T^{2} \) |
| 61 | \( 1 - 5.12T + 61T^{2} \) |
| 67 | \( 1 + 7.58T + 67T^{2} \) |
| 71 | \( 1 + 5.13T + 71T^{2} \) |
| 73 | \( 1 + 6.16T + 73T^{2} \) |
| 79 | \( 1 + 3.10T + 79T^{2} \) |
| 83 | \( 1 - 4.91T + 83T^{2} \) |
| 89 | \( 1 + 6.05T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−8.377305609108079802957671238057, −7.900234931407984976602003041547, −7.04698985862343469353794292551, −5.84040977368291669783003806189, −5.56782166202939644635478962271, −4.74682631538558091142916122891, −3.78748575743920643038300825793, −2.79760201233827928157364902869, −2.04478030465330577898286788933, −1.60754384117650729620789133006,
1.60754384117650729620789133006, 2.04478030465330577898286788933, 2.79760201233827928157364902869, 3.78748575743920643038300825793, 4.74682631538558091142916122891, 5.56782166202939644635478962271, 5.84040977368291669783003806189, 7.04698985862343469353794292551, 7.900234931407984976602003041547, 8.377305609108079802957671238057