L(s) = 1 | + 3-s + 7-s + 9-s − 5.05·11-s − 3.37·13-s − 7.18·17-s + 8.23·19-s + 21-s + 6.23·23-s + 27-s − 2·29-s − 4.62·31-s − 5.05·33-s + 4.85·37-s − 3.37·39-s − 3.37·41-s − 1.24·43-s + 49-s − 7.18·51-s − 4.62·53-s + 8.23·57-s − 11.6·59-s + 0.488·61-s + 63-s − 3.61·67-s + 6.23·69-s − 10.2·71-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.377·7-s + 0.333·9-s − 1.52·11-s − 0.936·13-s − 1.74·17-s + 1.88·19-s + 0.218·21-s + 1.30·23-s + 0.192·27-s − 0.371·29-s − 0.830·31-s − 0.879·33-s + 0.798·37-s − 0.540·39-s − 0.527·41-s − 0.189·43-s + 0.142·49-s − 1.00·51-s − 0.634·53-s + 1.09·57-s − 1.51·59-s + 0.0625·61-s + 0.125·63-s − 0.441·67-s + 0.750·69-s − 1.22·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 5.05T + 11T^{2} \) |
| 13 | \( 1 + 3.37T + 13T^{2} \) |
| 17 | \( 1 + 7.18T + 17T^{2} \) |
| 19 | \( 1 - 8.23T + 19T^{2} \) |
| 23 | \( 1 - 6.23T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 4.62T + 31T^{2} \) |
| 37 | \( 1 - 4.85T + 37T^{2} \) |
| 41 | \( 1 + 3.37T + 41T^{2} \) |
| 43 | \( 1 + 1.24T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 4.62T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 - 0.488T + 61T^{2} \) |
| 67 | \( 1 + 3.61T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 + 16.2T + 73T^{2} \) |
| 79 | \( 1 - 1.24T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 - 6.99T + 89T^{2} \) |
| 97 | \( 1 - 8.23T + 97T^{2} \) |
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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
−7.87172370645974286616975032757, −7.46795716431259077545486177873, −6.85212297635742462706175808599, −5.61871155673121720869025127149, −4.99464767735618546645588708149, −4.39193067773115724582865419534, −3.07631829127660821411662706522, −2.64323732261231479945707090360, −1.57310305169716197946147723672, 0,
1.57310305169716197946147723672, 2.64323732261231479945707090360, 3.07631829127660821411662706522, 4.39193067773115724582865419534, 4.99464767735618546645588708149, 5.61871155673121720869025127149, 6.85212297635742462706175808599, 7.46795716431259077545486177873, 7.87172370645974286616975032757