L(s) = 1 | + 3-s − 1.05·5-s + 3.84·7-s + 9-s − 2.99·11-s − 4.11·13-s − 1.05·15-s + 3.37·17-s − 3.86·19-s + 3.84·21-s − 23-s − 3.88·25-s + 27-s − 29-s + 1.19·31-s − 2.99·33-s − 4.05·35-s + 2.51·37-s − 4.11·39-s + 3.71·41-s − 5.77·43-s − 1.05·45-s + 6.95·47-s + 7.77·49-s + 3.37·51-s − 7.64·53-s + 3.15·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.471·5-s + 1.45·7-s + 0.333·9-s − 0.904·11-s − 1.14·13-s − 0.272·15-s + 0.817·17-s − 0.887·19-s + 0.838·21-s − 0.208·23-s − 0.777·25-s + 0.192·27-s − 0.185·29-s + 0.213·31-s − 0.521·33-s − 0.684·35-s + 0.412·37-s − 0.658·39-s + 0.579·41-s − 0.880·43-s − 0.157·45-s + 1.01·47-s + 1.11·49-s + 0.472·51-s − 1.04·53-s + 0.425·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
good | 5 | \( 1 + 1.05T + 5T^{2} \) |
| 7 | \( 1 - 3.84T + 7T^{2} \) |
| 11 | \( 1 + 2.99T + 11T^{2} \) |
| 13 | \( 1 + 4.11T + 13T^{2} \) |
| 17 | \( 1 - 3.37T + 17T^{2} \) |
| 19 | \( 1 + 3.86T + 19T^{2} \) |
| 31 | \( 1 - 1.19T + 31T^{2} \) |
| 37 | \( 1 - 2.51T + 37T^{2} \) |
| 41 | \( 1 - 3.71T + 41T^{2} \) |
| 43 | \( 1 + 5.77T + 43T^{2} \) |
| 47 | \( 1 - 6.95T + 47T^{2} \) |
| 53 | \( 1 + 7.64T + 53T^{2} \) |
| 59 | \( 1 - 8.37T + 59T^{2} \) |
| 61 | \( 1 + 9.45T + 61T^{2} \) |
| 67 | \( 1 + 5.84T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 + 8.75T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 + 4.62T + 89T^{2} \) |
| 97 | \( 1 - 5.25T + 97T^{2} \) |
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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
−7.66974132126711878247098171716, −7.17193612384849267130344645028, −5.99529005203086326251968410879, −5.25461969167223975027318240020, −4.58870331478598407023774343426, −4.06295711790939691339027970269, −2.96042468580183413574705856653, −2.26962779374633739531378940047, −1.44693462940829961205284083615, 0,
1.44693462940829961205284083615, 2.26962779374633739531378940047, 2.96042468580183413574705856653, 4.06295711790939691339027970269, 4.58870331478598407023774343426, 5.25461969167223975027318240020, 5.99529005203086326251968410879, 7.17193612384849267130344645028, 7.66974132126711878247098171716