L(s) = 1 | − 3·9-s − 11-s + 2·13-s − 6·17-s − 4·19-s + 4·23-s − 6·29-s + 8·31-s − 2·37-s + 2·41-s − 4·43-s − 12·47-s − 7·49-s − 2·53-s + 4·59-s + 10·61-s + 16·67-s − 8·71-s − 14·73-s − 8·79-s + 9·81-s + 4·83-s + 10·89-s − 10·97-s + 3·99-s + 101-s + 103-s + ⋯ |
L(s) = 1 | − 9-s − 0.301·11-s + 0.554·13-s − 1.45·17-s − 0.917·19-s + 0.834·23-s − 1.11·29-s + 1.43·31-s − 0.328·37-s + 0.312·41-s − 0.609·43-s − 1.75·47-s − 49-s − 0.274·53-s + 0.520·59-s + 1.28·61-s + 1.95·67-s − 0.949·71-s − 1.63·73-s − 0.900·79-s + 81-s + 0.439·83-s + 1.05·89-s − 1.01·97-s + 0.301·99-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.088628914\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.088628914\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.91022624964817, −15.17983219015281, −14.80243011511883, −14.26510568353667, −13.47334713241328, −13.14271820390713, −12.72114885513424, −11.75096141847157, −11.24589707648236, −11.07570315400737, −10.24092905422071, −9.649991993412616, −8.819375928911702, −8.546677207409666, −8.029423838814662, −7.103515319526206, −6.485229438804265, −6.088592828738763, −5.201913089263542, −4.702833518494879, −3.890483565950169, −3.127801226680374, −2.444029238442218, −1.681506284470787, −0.4244767051377569,
0.4244767051377569, 1.681506284470787, 2.444029238442218, 3.127801226680374, 3.890483565950169, 4.702833518494879, 5.201913089263542, 6.088592828738763, 6.485229438804265, 7.103515319526206, 8.029423838814662, 8.546677207409666, 8.819375928911702, 9.649991993412616, 10.24092905422071, 11.07570315400737, 11.24589707648236, 11.75096141847157, 12.72114885513424, 13.14271820390713, 13.47334713241328, 14.26510568353667, 14.80243011511883, 15.17983219015281, 15.91022624964817