L(s) = 1 | + 4·5-s − 3·9-s − 4·13-s + 8·17-s + 11·25-s + 10·29-s + 2·37-s + 8·41-s − 12·45-s + 14·53-s − 12·61-s − 16·65-s − 16·73-s + 9·81-s + 32·85-s − 16·89-s − 8·97-s + 20·101-s − 6·109-s − 14·113-s + 12·117-s + ⋯ |
L(s) = 1 | + 1.78·5-s − 9-s − 1.10·13-s + 1.94·17-s + 11/5·25-s + 1.85·29-s + 0.328·37-s + 1.24·41-s − 1.78·45-s + 1.92·53-s − 1.53·61-s − 1.98·65-s − 1.87·73-s + 81-s + 3.47·85-s − 1.69·89-s − 0.812·97-s + 1.99·101-s − 0.574·109-s − 1.31·113-s + 1.10·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.279726292\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.279726292\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−9.548865919223462988821068539485, −8.795935640120407431962303331374, −7.88132253194711275835728190267, −6.91533496238080223154082214854, −5.88267197317121816107420623197, −5.59269710161924499254999286275, −4.66389023955673654317656217449, −3.01558964211138052679351660845, −2.46224298486979636156274494161, −1.13590008709403894614989037078,
1.13590008709403894614989037078, 2.46224298486979636156274494161, 3.01558964211138052679351660845, 4.66389023955673654317656217449, 5.59269710161924499254999286275, 5.88267197317121816107420623197, 6.91533496238080223154082214854, 7.88132253194711275835728190267, 8.795935640120407431962303331374, 9.548865919223462988821068539485