L(s) = 1 | − 2.70·2-s − 3-s + 5.29·4-s + 2.21·5-s + 2.70·6-s − 2.11·7-s − 8.88·8-s + 9-s − 5.99·10-s + 11-s − 5.29·12-s − 2.52·13-s + 5.72·14-s − 2.21·15-s + 13.4·16-s + 1.86·17-s − 2.70·18-s − 1.62·19-s + 11.7·20-s + 2.11·21-s − 2.70·22-s + 0.232·23-s + 8.88·24-s − 0.0735·25-s + 6.81·26-s − 27-s − 11.2·28-s + ⋯ |
L(s) = 1 | − 1.90·2-s − 0.577·3-s + 2.64·4-s + 0.992·5-s + 1.10·6-s − 0.800·7-s − 3.14·8-s + 0.333·9-s − 1.89·10-s + 0.301·11-s − 1.52·12-s − 0.699·13-s + 1.52·14-s − 0.573·15-s + 3.35·16-s + 0.451·17-s − 0.636·18-s − 0.372·19-s + 2.62·20-s + 0.462·21-s − 0.575·22-s + 0.0484·23-s + 1.81·24-s − 0.0147·25-s + 1.33·26-s − 0.192·27-s − 2.11·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
good | 2 | \( 1 + 2.70T + 2T^{2} \) |
| 5 | \( 1 - 2.21T + 5T^{2} \) |
| 7 | \( 1 + 2.11T + 7T^{2} \) |
| 13 | \( 1 + 2.52T + 13T^{2} \) |
| 17 | \( 1 - 1.86T + 17T^{2} \) |
| 19 | \( 1 + 1.62T + 19T^{2} \) |
| 23 | \( 1 - 0.232T + 23T^{2} \) |
| 29 | \( 1 - 8.99T + 29T^{2} \) |
| 31 | \( 1 + 0.0529T + 31T^{2} \) |
| 37 | \( 1 + 1.12T + 37T^{2} \) |
| 41 | \( 1 + 8.95T + 41T^{2} \) |
| 43 | \( 1 + 8.83T + 43T^{2} \) |
| 47 | \( 1 + 8.15T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 + 1.89T + 59T^{2} \) |
| 67 | \( 1 + 1.07T + 67T^{2} \) |
| 71 | \( 1 - 5.42T + 71T^{2} \) |
| 73 | \( 1 + 8.03T + 73T^{2} \) |
| 79 | \( 1 + 6.67T + 79T^{2} \) |
| 83 | \( 1 - 3.06T + 83T^{2} \) |
| 89 | \( 1 - 5.40T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−8.874922598229526053124892826825, −8.208622536346452072552261229195, −7.15707717609761415513693795400, −6.57526995671263749289081490924, −6.07550643459120680797908633825, −5.04026660607631643662333639557, −3.28410550009955990852021099286, −2.27722901705640007760320447445, −1.31145948635138763594681064837, 0,
1.31145948635138763594681064837, 2.27722901705640007760320447445, 3.28410550009955990852021099286, 5.04026660607631643662333639557, 6.07550643459120680797908633825, 6.57526995671263749289081490924, 7.15707717609761415513693795400, 8.208622536346452072552261229195, 8.874922598229526053124892826825