| L(s) = 1 | + 2-s + 1.11·3-s + 4-s + 1.73·5-s + 1.11·6-s + 3.05·7-s + 8-s − 1.75·9-s + 1.73·10-s − 1.67·11-s + 1.11·12-s − 0.736·13-s + 3.05·14-s + 1.94·15-s + 16-s + 2.56·17-s − 1.75·18-s + 3.76·19-s + 1.73·20-s + 3.41·21-s − 1.67·22-s + 23-s + 1.11·24-s − 1.97·25-s − 0.736·26-s − 5.31·27-s + 3.05·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.645·3-s + 0.5·4-s + 0.777·5-s + 0.456·6-s + 1.15·7-s + 0.353·8-s − 0.583·9-s + 0.549·10-s − 0.504·11-s + 0.322·12-s − 0.204·13-s + 0.815·14-s + 0.501·15-s + 0.250·16-s + 0.621·17-s − 0.412·18-s + 0.863·19-s + 0.388·20-s + 0.744·21-s − 0.357·22-s + 0.208·23-s + 0.228·24-s − 0.395·25-s − 0.144·26-s − 1.02·27-s + 0.576·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.362544819\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.362544819\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 - 1.11T + 3T^{2} \) |
| 5 | \( 1 - 1.73T + 5T^{2} \) |
| 7 | \( 1 - 3.05T + 7T^{2} \) |
| 11 | \( 1 + 1.67T + 11T^{2} \) |
| 13 | \( 1 + 0.736T + 13T^{2} \) |
| 17 | \( 1 - 2.56T + 17T^{2} \) |
| 19 | \( 1 - 3.76T + 19T^{2} \) |
| 29 | \( 1 - 4.16T + 29T^{2} \) |
| 31 | \( 1 - 2.68T + 31T^{2} \) |
| 37 | \( 1 - 11.4T + 37T^{2} \) |
| 41 | \( 1 - 2.49T + 41T^{2} \) |
| 43 | \( 1 - 1.13T + 43T^{2} \) |
| 47 | \( 1 + 7.48T + 47T^{2} \) |
| 53 | \( 1 + 12.4T + 53T^{2} \) |
| 59 | \( 1 - 3.89T + 59T^{2} \) |
| 61 | \( 1 - 6.03T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 + 1.31T + 71T^{2} \) |
| 73 | \( 1 - 6.64T + 73T^{2} \) |
| 79 | \( 1 - 3.04T + 79T^{2} \) |
| 83 | \( 1 + 8.00T + 83T^{2} \) |
| 89 | \( 1 + 2.51T + 89T^{2} \) |
| 97 | \( 1 + 0.249T + 97T^{2} \) |
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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
−7.941110478760428140825733756402, −7.61650633366951978279115461709, −6.48260322164766185802317275459, −5.78735175536038524303715126786, −5.16968124254398139367024026489, −4.60528113106656528090083931875, −3.52942961376630758732853512710, −2.73394006439645032956024106398, −2.14580360153659515909310273377, −1.13506140929285708405929617453,
1.13506140929285708405929617453, 2.14580360153659515909310273377, 2.73394006439645032956024106398, 3.52942961376630758732853512710, 4.60528113106656528090083931875, 5.16968124254398139367024026489, 5.78735175536038524303715126786, 6.48260322164766185802317275459, 7.61650633366951978279115461709, 7.941110478760428140825733756402
