| L(s) = 1 | − 1.38·2-s − 0.110·3-s − 0.0682·4-s + 0.154·6-s − 1.71·7-s + 2.87·8-s − 2.98·9-s + 5.80·11-s + 0.00756·12-s − 1.25·13-s + 2.38·14-s − 3.85·16-s + 4.15·18-s + 2.82·19-s + 0.190·21-s − 8.06·22-s − 2.05·23-s − 0.318·24-s + 1.74·26-s + 0.663·27-s + 0.117·28-s + 5.20·29-s − 2.86·31-s − 0.386·32-s − 0.642·33-s + 0.203·36-s − 6.30·37-s + ⋯ |
| L(s) = 1 | − 0.982·2-s − 0.0639·3-s − 0.0341·4-s + 0.0628·6-s − 0.648·7-s + 1.01·8-s − 0.995·9-s + 1.74·11-s + 0.00218·12-s − 0.347·13-s + 0.637·14-s − 0.964·16-s + 0.978·18-s + 0.648·19-s + 0.0414·21-s − 1.71·22-s − 0.427·23-s − 0.0650·24-s + 0.341·26-s + 0.127·27-s + 0.0221·28-s + 0.966·29-s − 0.514·31-s − 0.0682·32-s − 0.111·33-s + 0.0339·36-s − 1.03·37-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 1.38T + 2T^{2} \) |
| 3 | \( 1 + 0.110T + 3T^{2} \) |
| 7 | \( 1 + 1.71T + 7T^{2} \) |
| 11 | \( 1 - 5.80T + 11T^{2} \) |
| 13 | \( 1 + 1.25T + 13T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 2.05T + 23T^{2} \) |
| 29 | \( 1 - 5.20T + 29T^{2} \) |
| 31 | \( 1 + 2.86T + 31T^{2} \) |
| 37 | \( 1 + 6.30T + 37T^{2} \) |
| 41 | \( 1 + 9.45T + 41T^{2} \) |
| 43 | \( 1 - 7.43T + 43T^{2} \) |
| 47 | \( 1 + 7.63T + 47T^{2} \) |
| 53 | \( 1 - 7.20T + 53T^{2} \) |
| 59 | \( 1 - 2.18T + 59T^{2} \) |
| 61 | \( 1 + 5.73T + 61T^{2} \) |
| 67 | \( 1 + 2.46T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 + 5.83T + 73T^{2} \) |
| 79 | \( 1 + 3.92T + 79T^{2} \) |
| 83 | \( 1 - 8.90T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 + 8.49T + 97T^{2} \) |
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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
−7.72547920134057545711154887866, −6.85538777617577319755027010520, −6.45994464519312555045106189317, −5.52608051158498444549863966841, −4.74672201078525534953935273381, −3.82136385797308640364254331911, −3.19000156591135821678996489191, −1.99267938716879523272461344494, −1.04113811833876441900761793834, 0,
1.04113811833876441900761793834, 1.99267938716879523272461344494, 3.19000156591135821678996489191, 3.82136385797308640364254331911, 4.74672201078525534953935273381, 5.52608051158498444549863966841, 6.45994464519312555045106189317, 6.85538777617577319755027010520, 7.72547920134057545711154887866
