L(s) = 1 | − 2-s + 2.38·3-s + 4-s − 2.03·5-s − 2.38·6-s − 1.02·7-s − 8-s + 2.66·9-s + 2.03·10-s − 1.20·11-s + 2.38·12-s − 2.91·13-s + 1.02·14-s − 4.85·15-s + 16-s + 0.372·17-s − 2.66·18-s + 2.99·19-s − 2.03·20-s − 2.45·21-s + 1.20·22-s + 0.188·23-s − 2.38·24-s − 0.841·25-s + 2.91·26-s − 0.789·27-s − 1.02·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.37·3-s + 0.5·4-s − 0.911·5-s − 0.971·6-s − 0.388·7-s − 0.353·8-s + 0.889·9-s + 0.644·10-s − 0.364·11-s + 0.687·12-s − 0.807·13-s + 0.275·14-s − 1.25·15-s + 0.250·16-s + 0.0902·17-s − 0.628·18-s + 0.686·19-s − 0.455·20-s − 0.534·21-s + 0.257·22-s + 0.0394·23-s − 0.485·24-s − 0.168·25-s + 0.570·26-s − 0.151·27-s − 0.194·28-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
\(L(1)\) |
\(\approx\) |
\(1.538041157\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.538041157\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.38T + 3T^{2} \) |
| 5 | \( 1 + 2.03T + 5T^{2} \) |
| 7 | \( 1 + 1.02T + 7T^{2} \) |
| 11 | \( 1 + 1.20T + 11T^{2} \) |
| 13 | \( 1 + 2.91T + 13T^{2} \) |
| 17 | \( 1 - 0.372T + 17T^{2} \) |
| 19 | \( 1 - 2.99T + 19T^{2} \) |
| 23 | \( 1 - 0.188T + 23T^{2} \) |
| 29 | \( 1 - 3.19T + 29T^{2} \) |
| 31 | \( 1 - 8.88T + 31T^{2} \) |
| 37 | \( 1 + 7.04T + 37T^{2} \) |
| 41 | \( 1 - 3.30T + 41T^{2} \) |
| 43 | \( 1 - 6.43T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 - 2.51T + 53T^{2} \) |
| 59 | \( 1 - 8.75T + 59T^{2} \) |
| 61 | \( 1 - 0.890T + 61T^{2} \) |
| 67 | \( 1 + 3.39T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 - 0.317T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 - 1.03T + 89T^{2} \) |
| 97 | \( 1 - 0.0889T + 97T^{2} \) |
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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
−8.421092274024869194676891018987, −7.79510506613082700038961352947, −7.40701455077445301150639226548, −6.61911357173565184384996862776, −5.48672700083764468112696179491, −4.41705787248677830492235936436, −3.56759866555864614670987361765, −2.86033640868646745935931837141, −2.19105264129027473393748918823, −0.71594944308404942315562889478,
0.71594944308404942315562889478, 2.19105264129027473393748918823, 2.86033640868646745935931837141, 3.56759866555864614670987361765, 4.41705787248677830492235936436, 5.48672700083764468112696179491, 6.61911357173565184384996862776, 7.40701455077445301150639226548, 7.79510506613082700038961352947, 8.421092274024869194676891018987