| L(s) = 1 | − 2.56·2-s + 3-s + 4.57·4-s − 3.57·5-s − 2.56·6-s + 2.10·7-s − 6.61·8-s + 9-s + 9.18·10-s + 3.23·11-s + 4.57·12-s − 6.86·13-s − 5.39·14-s − 3.57·15-s + 7.80·16-s + 2.22·17-s − 2.56·18-s + 5.62·19-s − 16.3·20-s + 2.10·21-s − 8.30·22-s − 4.53·23-s − 6.61·24-s + 7.81·25-s + 17.6·26-s + 27-s + 9.62·28-s + ⋯ |
| L(s) = 1 | − 1.81·2-s + 0.577·3-s + 2.28·4-s − 1.60·5-s − 1.04·6-s + 0.794·7-s − 2.33·8-s + 0.333·9-s + 2.90·10-s + 0.975·11-s + 1.32·12-s − 1.90·13-s − 1.44·14-s − 0.924·15-s + 1.95·16-s + 0.538·17-s − 0.604·18-s + 1.29·19-s − 3.66·20-s + 0.458·21-s − 1.77·22-s − 0.945·23-s − 1.35·24-s + 1.56·25-s + 3.45·26-s + 0.192·27-s + 1.81·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 669 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 669 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6269875197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6269875197\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 223 | \( 1 + T \) |
| good | 2 | \( 1 + 2.56T + 2T^{2} \) |
| 5 | \( 1 + 3.57T + 5T^{2} \) |
| 7 | \( 1 - 2.10T + 7T^{2} \) |
| 11 | \( 1 - 3.23T + 11T^{2} \) |
| 13 | \( 1 + 6.86T + 13T^{2} \) |
| 17 | \( 1 - 2.22T + 17T^{2} \) |
| 19 | \( 1 - 5.62T + 19T^{2} \) |
| 23 | \( 1 + 4.53T + 23T^{2} \) |
| 29 | \( 1 - 5.99T + 29T^{2} \) |
| 31 | \( 1 - 0.988T + 31T^{2} \) |
| 37 | \( 1 - 3.88T + 37T^{2} \) |
| 41 | \( 1 + 0.608T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 - 6.48T + 47T^{2} \) |
| 53 | \( 1 - 7.45T + 53T^{2} \) |
| 59 | \( 1 - 9.55T + 59T^{2} \) |
| 61 | \( 1 - 9.87T + 61T^{2} \) |
| 67 | \( 1 - 15.9T + 67T^{2} \) |
| 71 | \( 1 + 4.27T + 71T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 + 6.35T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 + 1.08T + 89T^{2} \) |
| 97 | \( 1 - 2.86T + 97T^{2} \) |
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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
−10.10656937571063554754350217664, −9.668785623025457294433016862080, −8.559973039359113107835050626396, −8.032923469307078764704901948942, −7.44148238822000803670217138209, −6.83045851857639147883539638382, −4.91850741042741853353053865654, −3.64629027434894103561746771784, −2.36060144095119581468109434054, −0.867375567347193073536131770028,
0.867375567347193073536131770028, 2.36060144095119581468109434054, 3.64629027434894103561746771784, 4.91850741042741853353053865654, 6.83045851857639147883539638382, 7.44148238822000803670217138209, 8.032923469307078764704901948942, 8.559973039359113107835050626396, 9.668785623025457294433016862080, 10.10656937571063554754350217664
