L(s) = 1 | + 2-s − 2.59·3-s + 4-s − 2.79·5-s − 2.59·6-s + 1.81·7-s + 8-s + 3.74·9-s − 2.79·10-s + 3.73·11-s − 2.59·12-s − 2.30·13-s + 1.81·14-s + 7.26·15-s + 16-s + 5.01·17-s + 3.74·18-s + 8.09·19-s − 2.79·20-s − 4.71·21-s + 3.73·22-s − 7.38·23-s − 2.59·24-s + 2.83·25-s − 2.30·26-s − 1.92·27-s + 1.81·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.49·3-s + 0.5·4-s − 1.25·5-s − 1.05·6-s + 0.685·7-s + 0.353·8-s + 1.24·9-s − 0.885·10-s + 1.12·11-s − 0.749·12-s − 0.639·13-s + 0.484·14-s + 1.87·15-s + 0.250·16-s + 1.21·17-s + 0.881·18-s + 1.85·19-s − 0.625·20-s − 1.02·21-s + 0.796·22-s − 1.54·23-s − 0.529·24-s + 0.566·25-s − 0.452·26-s − 0.370·27-s + 0.342·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.471967564\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.471967564\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 2017 | \( 1 + T \) |
good | 3 | \( 1 + 2.59T + 3T^{2} \) |
| 5 | \( 1 + 2.79T + 5T^{2} \) |
| 7 | \( 1 - 1.81T + 7T^{2} \) |
| 11 | \( 1 - 3.73T + 11T^{2} \) |
| 13 | \( 1 + 2.30T + 13T^{2} \) |
| 17 | \( 1 - 5.01T + 17T^{2} \) |
| 19 | \( 1 - 8.09T + 19T^{2} \) |
| 23 | \( 1 + 7.38T + 23T^{2} \) |
| 29 | \( 1 + 6.39T + 29T^{2} \) |
| 31 | \( 1 + 7.51T + 31T^{2} \) |
| 37 | \( 1 - 0.201T + 37T^{2} \) |
| 41 | \( 1 - 4.54T + 41T^{2} \) |
| 43 | \( 1 - 5.72T + 43T^{2} \) |
| 47 | \( 1 + 6.67T + 47T^{2} \) |
| 53 | \( 1 + 8.13T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 - 5.99T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 - 2.14T + 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 - 8.15T + 83T^{2} \) |
| 89 | \( 1 - 1.40T + 89T^{2} \) |
| 97 | \( 1 + 0.714T + 97T^{2} \) |
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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
−7.944976384691168031683662207823, −7.60849364374454108158226969719, −6.94793710076807915737262050025, −5.98159206049428361067012477315, −5.41853491101077424802818383393, −4.80739066030811062274050956055, −3.92630716601295896523412688575, −3.44834794378102168318408432976, −1.75743663589185745404700762578, −0.69766329653105864453353732483,
0.69766329653105864453353732483, 1.75743663589185745404700762578, 3.44834794378102168318408432976, 3.92630716601295896523412688575, 4.80739066030811062274050956055, 5.41853491101077424802818383393, 5.98159206049428361067012477315, 6.94793710076807915737262050025, 7.60849364374454108158226969719, 7.944976384691168031683662207823