| L(s) = 1 | + 2.64·2-s − 3-s + 5.01·4-s + 5-s − 2.64·6-s − 0.530·7-s + 8.00·8-s + 9-s + 2.64·10-s + 3.81·11-s − 5.01·12-s + 0.0914·13-s − 1.40·14-s − 15-s + 11.1·16-s + 7.19·17-s + 2.64·18-s + 0.705·19-s + 5.01·20-s + 0.530·21-s + 10.1·22-s − 3.42·23-s − 8.00·24-s + 25-s + 0.242·26-s − 27-s − 2.66·28-s + ⋯ |
| L(s) = 1 | + 1.87·2-s − 0.577·3-s + 2.50·4-s + 0.447·5-s − 1.08·6-s − 0.200·7-s + 2.82·8-s + 0.333·9-s + 0.837·10-s + 1.15·11-s − 1.44·12-s + 0.0253·13-s − 0.375·14-s − 0.258·15-s + 2.78·16-s + 1.74·17-s + 0.624·18-s + 0.161·19-s + 1.12·20-s + 0.115·21-s + 2.15·22-s − 0.714·23-s − 1.63·24-s + 0.200·25-s + 0.0475·26-s − 0.192·27-s − 0.503·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.195183053\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.195183053\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 - 2.64T + 2T^{2} \) |
| 7 | \( 1 + 0.530T + 7T^{2} \) |
| 11 | \( 1 - 3.81T + 11T^{2} \) |
| 13 | \( 1 - 0.0914T + 13T^{2} \) |
| 17 | \( 1 - 7.19T + 17T^{2} \) |
| 19 | \( 1 - 0.705T + 19T^{2} \) |
| 23 | \( 1 + 3.42T + 23T^{2} \) |
| 29 | \( 1 - 4.28T + 29T^{2} \) |
| 31 | \( 1 + 4.83T + 31T^{2} \) |
| 37 | \( 1 + 8.28T + 37T^{2} \) |
| 41 | \( 1 + 2.75T + 41T^{2} \) |
| 43 | \( 1 - 7.14T + 43T^{2} \) |
| 47 | \( 1 + 6.22T + 47T^{2} \) |
| 53 | \( 1 + 6.85T + 53T^{2} \) |
| 59 | \( 1 - 8.63T + 59T^{2} \) |
| 61 | \( 1 - 14.6T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 + 5.64T + 73T^{2} \) |
| 79 | \( 1 + 7.02T + 79T^{2} \) |
| 83 | \( 1 + 14.8T + 83T^{2} \) |
| 89 | \( 1 - 8.37T + 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−7.65431729148166852480976345923, −6.94168161997303957544063805039, −6.41357202275456263329415796067, −5.74475565766709770461703386026, −5.31016275670169983009104152065, −4.51283168786089481315144530166, −3.67128236546290945593369278940, −3.22580608891672407134206140985, −2.02321522415968123277895083726, −1.22058629118473109071072562472,
1.22058629118473109071072562472, 2.02321522415968123277895083726, 3.22580608891672407134206140985, 3.67128236546290945593369278940, 4.51283168786089481315144530166, 5.31016275670169983009104152065, 5.74475565766709770461703386026, 6.41357202275456263329415796067, 6.94168161997303957544063805039, 7.65431729148166852480976345923
