| L(s) = 1 | + 2-s − 3-s + 4-s + 1.40·5-s − 6-s + 4.86·7-s + 8-s + 9-s + 1.40·10-s − 2.33·11-s − 12-s − 13-s + 4.86·14-s − 1.40·15-s + 16-s + 5.38·17-s + 18-s + 5.46·19-s + 1.40·20-s − 4.86·21-s − 2.33·22-s − 1.00·23-s − 24-s − 3.01·25-s − 26-s − 27-s + 4.86·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.630·5-s − 0.408·6-s + 1.84·7-s + 0.353·8-s + 0.333·9-s + 0.445·10-s − 0.702·11-s − 0.288·12-s − 0.277·13-s + 1.30·14-s − 0.364·15-s + 0.250·16-s + 1.30·17-s + 0.235·18-s + 1.25·19-s + 0.315·20-s − 1.06·21-s − 0.496·22-s − 0.210·23-s − 0.204·24-s − 0.602·25-s − 0.196·26-s − 0.192·27-s + 0.920·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.227687917\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.227687917\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 5 | \( 1 - 1.40T + 5T^{2} \) |
| 7 | \( 1 - 4.86T + 7T^{2} \) |
| 11 | \( 1 + 2.33T + 11T^{2} \) |
| 17 | \( 1 - 5.38T + 17T^{2} \) |
| 19 | \( 1 - 5.46T + 19T^{2} \) |
| 23 | \( 1 + 1.00T + 23T^{2} \) |
| 29 | \( 1 + 2.01T + 29T^{2} \) |
| 31 | \( 1 - 0.445T + 31T^{2} \) |
| 37 | \( 1 - 9.09T + 37T^{2} \) |
| 41 | \( 1 + 0.952T + 41T^{2} \) |
| 43 | \( 1 + 3.56T + 43T^{2} \) |
| 47 | \( 1 - 0.884T + 47T^{2} \) |
| 53 | \( 1 + 7.50T + 53T^{2} \) |
| 59 | \( 1 - 4.52T + 59T^{2} \) |
| 61 | \( 1 - 6.62T + 61T^{2} \) |
| 67 | \( 1 - 2.00T + 67T^{2} \) |
| 71 | \( 1 + 9.19T + 71T^{2} \) |
| 73 | \( 1 + 6.35T + 73T^{2} \) |
| 79 | \( 1 - 3.70T + 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 - 2.48T + 89T^{2} \) |
| 97 | \( 1 - 6.09T + 97T^{2} \) |
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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
−7.79346771568039562963108542006, −7.22877357029536662235122934371, −6.12615133369181705924278410488, −5.52746464147130011748114649026, −5.15920712465760717574807027794, −4.57796076912746175806474677785, −3.63816386385386455446600102084, −2.59752213003222899023965579567, −1.77507258606155298874159220304, −1.02370677126666903347360097673,
1.02370677126666903347360097673, 1.77507258606155298874159220304, 2.59752213003222899023965579567, 3.63816386385386455446600102084, 4.57796076912746175806474677785, 5.15920712465760717574807027794, 5.52746464147130011748114649026, 6.12615133369181705924278410488, 7.22877357029536662235122934371, 7.79346771568039562963108542006
