L(s) = 1 | + 2-s − 3-s + 4-s − 0.254·5-s − 6-s + 2.89·7-s + 8-s + 9-s − 0.254·10-s − 3.38·11-s − 12-s + 13-s + 2.89·14-s + 0.254·15-s + 16-s − 2.25·17-s + 18-s − 4.71·19-s − 0.254·20-s − 2.89·21-s − 3.38·22-s + 7.05·23-s − 24-s − 4.93·25-s + 26-s − 27-s + 2.89·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.114·5-s − 0.408·6-s + 1.09·7-s + 0.353·8-s + 0.333·9-s − 0.0806·10-s − 1.02·11-s − 0.288·12-s + 0.277·13-s + 0.773·14-s + 0.0658·15-s + 0.250·16-s − 0.546·17-s + 0.235·18-s − 1.08·19-s − 0.0570·20-s − 0.631·21-s − 0.721·22-s + 1.47·23-s − 0.204·24-s − 0.986·25-s + 0.196·26-s − 0.192·27-s + 0.547·28-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 | \( 1 + 0.254T + 5T^{2} \) |
| 7 | \( 1 - 2.89T + 7T^{2} \) |
| 11 | \( 1 + 3.38T + 11T^{2} \) |
| 17 | \( 1 + 2.25T + 17T^{2} \) |
| 19 | \( 1 + 4.71T + 19T^{2} \) |
| 23 | \( 1 - 7.05T + 23T^{2} \) |
| 29 | \( 1 + 6.73T + 29T^{2} \) |
| 31 | \( 1 + 3.07T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + 4.02T + 41T^{2} \) |
| 43 | \( 1 + 9.59T + 43T^{2} \) |
| 47 | \( 1 - 8.51T + 47T^{2} \) |
| 53 | \( 1 + 0.171T + 53T^{2} \) |
| 59 | \( 1 - 7.58T + 59T^{2} \) |
| 61 | \( 1 + 1.83T + 61T^{2} \) |
| 67 | \( 1 + 5.17T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 + 8.40T + 79T^{2} \) |
| 83 | \( 1 + 15.9T + 83T^{2} \) |
| 89 | \( 1 - 3.35T + 89T^{2} \) |
| 97 | \( 1 + 8.53T + 97T^{2} \) |
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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
−7.33680619353275366397192659788, −6.81791708952590709673756260539, −5.70831716953476988538899095096, −5.56128043547385342597488321336, −4.52138931560740128617948909480, −4.30584563470544801432230529115, −3.14635970046549897597116141163, −2.21887292966317093482488498579, −1.45115796584244731893326838365, 0,
1.45115796584244731893326838365, 2.21887292966317093482488498579, 3.14635970046549897597116141163, 4.30584563470544801432230529115, 4.52138931560740128617948909480, 5.56128043547385342597488321336, 5.70831716953476988538899095096, 6.81791708952590709673756260539, 7.33680619353275366397192659788