L(s) = 1 | − 2-s + 3-s + 4-s + 1.70·5-s − 6-s + 3.64·7-s − 8-s + 9-s − 1.70·10-s − 3.90·11-s + 12-s + 4.84·13-s − 3.64·14-s + 1.70·15-s + 16-s − 0.120·17-s − 18-s + 19-s + 1.70·20-s + 3.64·21-s + 3.90·22-s + 5.23·23-s − 24-s − 2.08·25-s − 4.84·26-s + 27-s + 3.64·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.763·5-s − 0.408·6-s + 1.37·7-s − 0.353·8-s + 0.333·9-s − 0.540·10-s − 1.17·11-s + 0.288·12-s + 1.34·13-s − 0.973·14-s + 0.441·15-s + 0.250·16-s − 0.0292·17-s − 0.235·18-s + 0.229·19-s + 0.381·20-s + 0.795·21-s + 0.832·22-s + 1.09·23-s − 0.204·24-s − 0.416·25-s − 0.949·26-s + 0.192·27-s + 0.688·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.662593530\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.662593530\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 - T \) |
good | 5 | \( 1 - 1.70T + 5T^{2} \) |
| 7 | \( 1 - 3.64T + 7T^{2} \) |
| 11 | \( 1 + 3.90T + 11T^{2} \) |
| 13 | \( 1 - 4.84T + 13T^{2} \) |
| 17 | \( 1 + 0.120T + 17T^{2} \) |
| 23 | \( 1 - 5.23T + 23T^{2} \) |
| 29 | \( 1 + 10.6T + 29T^{2} \) |
| 31 | \( 1 - 5.67T + 31T^{2} \) |
| 37 | \( 1 - 2.82T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 + 3.23T + 43T^{2} \) |
| 47 | \( 1 + 3.45T + 47T^{2} \) |
| 59 | \( 1 + 2.00T + 59T^{2} \) |
| 61 | \( 1 - 8.20T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 + 5.50T + 71T^{2} \) |
| 73 | \( 1 + 2.35T + 73T^{2} \) |
| 79 | \( 1 + 5.65T + 79T^{2} \) |
| 83 | \( 1 - 7.40T + 83T^{2} \) |
| 89 | \( 1 + 6.28T + 89T^{2} \) |
| 97 | \( 1 - 19.0T + 97T^{2} \) |
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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
−7.995242334826856508171838535729, −7.75991460456971439222235316479, −6.85164021162898550504329176880, −5.86152516373512079435394875790, −5.37975800645970996503778235317, −4.47356009584720669601542807997, −3.45519252055133095972350910451, −2.47515769865778637880264385463, −1.82571248038550808222269809654, −0.987510750219154908485646926360,
0.987510750219154908485646926360, 1.82571248038550808222269809654, 2.47515769865778637880264385463, 3.45519252055133095972350910451, 4.47356009584720669601542807997, 5.37975800645970996503778235317, 5.86152516373512079435394875790, 6.85164021162898550504329176880, 7.75991460456971439222235316479, 7.995242334826856508171838535729