L(s) = 1 | − 1.31·2-s + 3-s − 0.265·4-s + 5-s − 1.31·6-s + 1.83·7-s + 2.98·8-s + 9-s − 1.31·10-s − 3.89·11-s − 0.265·12-s + 13-s − 2.41·14-s + 15-s − 3.39·16-s − 5.72·17-s − 1.31·18-s + 3.17·19-s − 0.265·20-s + 1.83·21-s + 5.12·22-s + 4.42·23-s + 2.98·24-s + 25-s − 1.31·26-s + 27-s − 0.487·28-s + ⋯ |
L(s) = 1 | − 0.931·2-s + 0.577·3-s − 0.132·4-s + 0.447·5-s − 0.537·6-s + 0.692·7-s + 1.05·8-s + 0.333·9-s − 0.416·10-s − 1.17·11-s − 0.0766·12-s + 0.277·13-s − 0.645·14-s + 0.258·15-s − 0.849·16-s − 1.38·17-s − 0.310·18-s + 0.727·19-s − 0.0594·20-s + 0.400·21-s + 1.09·22-s + 0.923·23-s + 0.609·24-s + 0.200·25-s − 0.258·26-s + 0.192·27-s − 0.0920·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 - T \) |
good | 2 | \( 1 + 1.31T + 2T^{2} \) |
| 7 | \( 1 - 1.83T + 7T^{2} \) |
| 11 | \( 1 + 3.89T + 11T^{2} \) |
| 17 | \( 1 + 5.72T + 17T^{2} \) |
| 19 | \( 1 - 3.17T + 19T^{2} \) |
| 23 | \( 1 - 4.42T + 23T^{2} \) |
| 29 | \( 1 + 7.39T + 29T^{2} \) |
| 37 | \( 1 + 4.72T + 37T^{2} \) |
| 41 | \( 1 + 7.37T + 41T^{2} \) |
| 43 | \( 1 - 3.88T + 43T^{2} \) |
| 47 | \( 1 - 7.26T + 47T^{2} \) |
| 53 | \( 1 + 3.54T + 53T^{2} \) |
| 59 | \( 1 + 7.51T + 59T^{2} \) |
| 61 | \( 1 + 7.62T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 + 0.175T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 - 5.79T + 79T^{2} \) |
| 83 | \( 1 - 4.95T + 83T^{2} \) |
| 89 | \( 1 - 0.109T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−7.84788819375662747879968573180, −7.37365392525110414060946048733, −6.55746167473826154785832268117, −5.36401895479931234933903931891, −4.92742204722006975973282750550, −4.08444509208124542595434624048, −3.00085759467330370623096853054, −2.07937277850727288801110376985, −1.36569949871766833189691658110, 0,
1.36569949871766833189691658110, 2.07937277850727288801110376985, 3.00085759467330370623096853054, 4.08444509208124542595434624048, 4.92742204722006975973282750550, 5.36401895479931234933903931891, 6.55746167473826154785832268117, 7.37365392525110414060946048733, 7.84788819375662747879968573180