L(s) = 1 | + 2-s − 3-s + 4-s − 0.760·5-s − 6-s − 1.62·7-s + 8-s + 9-s − 0.760·10-s + 3.63·11-s − 12-s + 6.76·13-s − 1.62·14-s + 0.760·15-s + 16-s − 17-s + 18-s + 2.10·19-s − 0.760·20-s + 1.62·21-s + 3.63·22-s + 7.89·23-s − 24-s − 4.42·25-s + 6.76·26-s − 27-s − 1.62·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.340·5-s − 0.408·6-s − 0.615·7-s + 0.353·8-s + 0.333·9-s − 0.240·10-s + 1.09·11-s − 0.288·12-s + 1.87·13-s − 0.434·14-s + 0.196·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s + 0.483·19-s − 0.170·20-s + 0.355·21-s + 0.773·22-s + 1.64·23-s − 0.204·24-s − 0.884·25-s + 1.32·26-s − 0.192·27-s − 0.307·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.754213692\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.754213692\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
good | 5 | \( 1 + 0.760T + 5T^{2} \) |
| 7 | \( 1 + 1.62T + 7T^{2} \) |
| 11 | \( 1 - 3.63T + 11T^{2} \) |
| 13 | \( 1 - 6.76T + 13T^{2} \) |
| 19 | \( 1 - 2.10T + 19T^{2} \) |
| 23 | \( 1 - 7.89T + 23T^{2} \) |
| 29 | \( 1 + 2.99T + 29T^{2} \) |
| 31 | \( 1 + 5.23T + 31T^{2} \) |
| 37 | \( 1 - 5.67T + 37T^{2} \) |
| 41 | \( 1 - 4.04T + 41T^{2} \) |
| 43 | \( 1 + 6.63T + 43T^{2} \) |
| 47 | \( 1 + 1.45T + 47T^{2} \) |
| 53 | \( 1 - 1.88T + 53T^{2} \) |
| 61 | \( 1 - 13.6T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 - 6.36T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 - 1.53T + 83T^{2} \) |
| 89 | \( 1 - 3.89T + 89T^{2} \) |
| 97 | \( 1 - 15.9T + 97T^{2} \) |
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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
−7.914537806997406680186173226199, −7.09644031687614684882746157736, −6.49661688364660256527186217009, −6.00149892136812385961283899071, −5.29573041597626413195969368930, −4.32145097915827910746162394166, −3.68717241947630413673568206723, −3.18795448617559548443531316696, −1.74940356495530072567829678744, −0.854593226449390796254641889613,
0.854593226449390796254641889613, 1.74940356495530072567829678744, 3.18795448617559548443531316696, 3.68717241947630413673568206723, 4.32145097915827910746162394166, 5.29573041597626413195969368930, 6.00149892136812385961283899071, 6.49661688364660256527186217009, 7.09644031687614684882746157736, 7.914537806997406680186173226199