| L(s) = 1 | − 2-s − 3-s + 4-s + 2.66·5-s + 6-s + 2.36·7-s − 8-s + 9-s − 2.66·10-s + 2.99·11-s − 12-s − 13-s − 2.36·14-s − 2.66·15-s + 16-s − 5.10·17-s − 18-s − 4.97·19-s + 2.66·20-s − 2.36·21-s − 2.99·22-s + 8.19·23-s + 24-s + 2.10·25-s + 26-s − 27-s + 2.36·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.19·5-s + 0.408·6-s + 0.895·7-s − 0.353·8-s + 0.333·9-s − 0.842·10-s + 0.901·11-s − 0.288·12-s − 0.277·13-s − 0.633·14-s − 0.688·15-s + 0.250·16-s − 1.23·17-s − 0.235·18-s − 1.14·19-s + 0.595·20-s − 0.516·21-s − 0.637·22-s + 1.70·23-s + 0.204·24-s + 0.420·25-s + 0.196·26-s − 0.192·27-s + 0.447·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 5 | \( 1 - 2.66T + 5T^{2} \) |
| 7 | \( 1 - 2.36T + 7T^{2} \) |
| 11 | \( 1 - 2.99T + 11T^{2} \) |
| 17 | \( 1 + 5.10T + 17T^{2} \) |
| 19 | \( 1 + 4.97T + 19T^{2} \) |
| 23 | \( 1 - 8.19T + 23T^{2} \) |
| 29 | \( 1 + 8.89T + 29T^{2} \) |
| 31 | \( 1 - 2.96T + 31T^{2} \) |
| 37 | \( 1 + 7.50T + 37T^{2} \) |
| 41 | \( 1 + 5.29T + 41T^{2} \) |
| 43 | \( 1 - 5.03T + 43T^{2} \) |
| 47 | \( 1 + 6.68T + 47T^{2} \) |
| 53 | \( 1 + 9.57T + 53T^{2} \) |
| 59 | \( 1 - 4.13T + 59T^{2} \) |
| 61 | \( 1 + 0.907T + 61T^{2} \) |
| 67 | \( 1 + 3.14T + 67T^{2} \) |
| 71 | \( 1 + 9.42T + 71T^{2} \) |
| 73 | \( 1 - 1.53T + 73T^{2} \) |
| 79 | \( 1 + 16.6T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 + 5.63T + 97T^{2} \) |
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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
−7.36231914217911530392235364072, −6.71617249049254199167514993659, −6.30682236516171976583547907977, −5.44675761192350496468735362773, −4.84840891567222161096802322769, −4.04685072927775339468812261490, −2.78847784130197712027339842667, −1.77138244117482798440757826176, −1.50675007339327076443603491070, 0,
1.50675007339327076443603491070, 1.77138244117482798440757826176, 2.78847784130197712027339842667, 4.04685072927775339468812261490, 4.84840891567222161096802322769, 5.44675761192350496468735362773, 6.30682236516171976583547907977, 6.71617249049254199167514993659, 7.36231914217911530392235364072
