| L(s) = 1 | + 2.38·2-s + 3.15·3-s + 3.69·4-s + 7.52·6-s − 0.219·7-s + 4.04·8-s + 6.95·9-s − 0.524·11-s + 11.6·12-s + 1.96·13-s − 0.524·14-s + 2.25·16-s + 16.5·18-s + 4·19-s − 0.693·21-s − 1.25·22-s + 0.372·23-s + 12.7·24-s + 4.69·26-s + 12.4·27-s − 0.812·28-s − 7.00·29-s − 2.92·31-s − 2.69·32-s − 1.65·33-s + 25.6·36-s + 5.71·37-s + ⋯ |
| L(s) = 1 | + 1.68·2-s + 1.82·3-s + 1.84·4-s + 3.07·6-s − 0.0831·7-s + 1.42·8-s + 2.31·9-s − 0.158·11-s + 3.36·12-s + 0.545·13-s − 0.140·14-s + 0.564·16-s + 3.90·18-s + 0.917·19-s − 0.151·21-s − 0.267·22-s + 0.0777·23-s + 2.60·24-s + 0.920·26-s + 2.39·27-s − 0.153·28-s − 1.30·29-s − 0.524·31-s − 0.476·32-s − 0.288·33-s + 4.27·36-s + 0.939·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.83978360\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.83978360\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 2.38T + 2T^{2} \) |
| 3 | \( 1 - 3.15T + 3T^{2} \) |
| 7 | \( 1 + 0.219T + 7T^{2} \) |
| 11 | \( 1 + 0.524T + 11T^{2} \) |
| 13 | \( 1 - 1.96T + 13T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 0.372T + 23T^{2} \) |
| 29 | \( 1 + 7.00T + 29T^{2} \) |
| 31 | \( 1 + 2.92T + 31T^{2} \) |
| 37 | \( 1 - 5.71T + 37T^{2} \) |
| 41 | \( 1 - 0.797T + 41T^{2} \) |
| 43 | \( 1 + 2.49T + 43T^{2} \) |
| 47 | \( 1 - 6.73T + 47T^{2} \) |
| 53 | \( 1 + 5.92T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 5.65T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 - 7.16T + 71T^{2} \) |
| 73 | \( 1 - 1.18T + 73T^{2} \) |
| 79 | \( 1 + 6.73T + 79T^{2} \) |
| 83 | \( 1 + 6.11T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 + 9.21T + 97T^{2} \) |
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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
−7.76306721342517232646733874169, −7.22638747339993643161256467745, −6.51027853843554156942855868698, −5.63752796777087701024921695056, −4.91395520911893385159438043489, −4.01507452199338953154732471290, −3.63073600550792145683244077420, −2.96037581184454504088304889253, −2.30510210110262100409794332477, −1.45153724093597613979517661278,
1.45153724093597613979517661278, 2.30510210110262100409794332477, 2.96037581184454504088304889253, 3.63073600550792145683244077420, 4.01507452199338953154732471290, 4.91395520911893385159438043489, 5.63752796777087701024921695056, 6.51027853843554156942855868698, 7.22638747339993643161256467745, 7.76306721342517232646733874169
