| L(s) = 1 | + 2-s + 3-s + 4-s + 2.72·5-s + 6-s − 0.673·7-s + 8-s + 9-s + 2.72·10-s − 11-s + 12-s + 3.59·13-s − 0.673·14-s + 2.72·15-s + 16-s + 4.25·17-s + 18-s + 5.82·19-s + 2.72·20-s − 0.673·21-s − 22-s − 2.69·23-s + 24-s + 2.44·25-s + 3.59·26-s + 27-s − 0.673·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.22·5-s + 0.408·6-s − 0.254·7-s + 0.353·8-s + 0.333·9-s + 0.863·10-s − 0.301·11-s + 0.288·12-s + 0.997·13-s − 0.180·14-s + 0.704·15-s + 0.250·16-s + 1.03·17-s + 0.235·18-s + 1.33·19-s + 0.610·20-s − 0.147·21-s − 0.213·22-s − 0.561·23-s + 0.204·24-s + 0.489·25-s + 0.705·26-s + 0.192·27-s − 0.127·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.108098533\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.108098533\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 5 | \( 1 - 2.72T + 5T^{2} \) |
| 7 | \( 1 + 0.673T + 7T^{2} \) |
| 13 | \( 1 - 3.59T + 13T^{2} \) |
| 17 | \( 1 - 4.25T + 17T^{2} \) |
| 19 | \( 1 - 5.82T + 19T^{2} \) |
| 23 | \( 1 + 2.69T + 23T^{2} \) |
| 29 | \( 1 + 7.00T + 29T^{2} \) |
| 31 | \( 1 - 1.96T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 + 3.88T + 41T^{2} \) |
| 43 | \( 1 + 9.14T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 + 3.56T + 53T^{2} \) |
| 59 | \( 1 - 3.91T + 59T^{2} \) |
| 67 | \( 1 - 6.94T + 67T^{2} \) |
| 71 | \( 1 - 0.365T + 71T^{2} \) |
| 73 | \( 1 + 4.95T + 73T^{2} \) |
| 79 | \( 1 + 2.75T + 79T^{2} \) |
| 83 | \( 1 + 0.156T + 83T^{2} \) |
| 89 | \( 1 - 16.9T + 89T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
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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
−8.274948552245636030201278965811, −7.77280900116361454928813187866, −6.80782937161184596462830445695, −6.05857923493278637977675838446, −5.55936310641252305977846996791, −4.77781778679530734669593646413, −3.56779305901228411941021536853, −3.14984265359743266265843895483, −2.06456731944300826561798943850, −1.28528413614215449769657897313,
1.28528413614215449769657897313, 2.06456731944300826561798943850, 3.14984265359743266265843895483, 3.56779305901228411941021536853, 4.77781778679530734669593646413, 5.55936310641252305977846996791, 6.05857923493278637977675838446, 6.80782937161184596462830445695, 7.77280900116361454928813187866, 8.274948552245636030201278965811
