| L(s) = 1 | + 2-s + 0.500·3-s + 4-s − 1.03·5-s + 0.500·6-s + 3.13·7-s + 8-s − 2.74·9-s − 1.03·10-s + 11-s + 0.500·12-s − 4.91·13-s + 3.13·14-s − 0.518·15-s + 16-s + 1.18·17-s − 2.74·18-s − 5.27·19-s − 1.03·20-s + 1.57·21-s + 22-s − 9.20·23-s + 0.500·24-s − 3.92·25-s − 4.91·26-s − 2.87·27-s + 3.13·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.288·3-s + 0.5·4-s − 0.463·5-s + 0.204·6-s + 1.18·7-s + 0.353·8-s − 0.916·9-s − 0.327·10-s + 0.301·11-s + 0.144·12-s − 1.36·13-s + 0.838·14-s − 0.133·15-s + 0.250·16-s + 0.287·17-s − 0.648·18-s − 1.21·19-s − 0.231·20-s + 0.342·21-s + 0.213·22-s − 1.92·23-s + 0.102·24-s − 0.785·25-s − 0.963·26-s − 0.553·27-s + 0.593·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 + T \) |
| good | 3 | \( 1 - 0.500T + 3T^{2} \) |
| 5 | \( 1 + 1.03T + 5T^{2} \) |
| 7 | \( 1 - 3.13T + 7T^{2} \) |
| 13 | \( 1 + 4.91T + 13T^{2} \) |
| 17 | \( 1 - 1.18T + 17T^{2} \) |
| 19 | \( 1 + 5.27T + 19T^{2} \) |
| 23 | \( 1 + 9.20T + 23T^{2} \) |
| 29 | \( 1 - 2.82T + 29T^{2} \) |
| 31 | \( 1 + 1.66T + 31T^{2} \) |
| 37 | \( 1 - 12.0T + 37T^{2} \) |
| 41 | \( 1 + 6.75T + 41T^{2} \) |
| 43 | \( 1 - 1.17T + 43T^{2} \) |
| 47 | \( 1 + 6.36T + 47T^{2} \) |
| 53 | \( 1 + 9.47T + 53T^{2} \) |
| 59 | \( 1 + 1.23T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 - 1.09T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 + 7.78T + 73T^{2} \) |
| 79 | \( 1 + 0.496T + 79T^{2} \) |
| 83 | \( 1 - 5.98T + 83T^{2} \) |
| 89 | \( 1 - 2.48T + 89T^{2} \) |
| 97 | \( 1 + 16.8T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.047036516865139977678921163326, −7.46826142519508088820354475251, −6.35664701947043452573086886393, −5.79337642821938414150876749438, −4.78661728926962735121806134333, −4.38610624069680621935438629535, −3.46633988880621556625644635025, −2.43394900406766749581138496561, −1.81079894920027954095094012425, 0,
1.81079894920027954095094012425, 2.43394900406766749581138496561, 3.46633988880621556625644635025, 4.38610624069680621935438629535, 4.78661728926962735121806134333, 5.79337642821938414150876749438, 6.35664701947043452573086886393, 7.46826142519508088820354475251, 8.047036516865139977678921163326
