L(s) = 1 | − 7-s + 2·11-s + 6·13-s − 4·17-s + 4·19-s − 2·23-s + 2·29-s − 2·37-s − 4·43-s − 12·47-s + 49-s − 6·53-s − 8·59-s + 6·61-s − 8·67-s + 14·71-s + 2·73-s − 2·77-s − 12·79-s + 4·83-s − 6·91-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | − 0.377·7-s + 0.603·11-s + 1.66·13-s − 0.970·17-s + 0.917·19-s − 0.417·23-s + 0.371·29-s − 0.328·37-s − 0.609·43-s − 1.75·47-s + 1/7·49-s − 0.824·53-s − 1.04·59-s + 0.768·61-s − 0.977·67-s + 1.66·71-s + 0.234·73-s − 0.227·77-s − 1.35·79-s + 0.439·83-s − 0.628·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.88541683638899, −15.13656378010121, −14.54772231600569, −13.85941037768347, −13.53640869231775, −13.09058458349636, −12.35412752101343, −11.85452918000077, −11.14755537879427, −10.97610983071725, −10.12455145454971, −9.541820128485428, −9.098122557296207, −8.372135528135169, −8.086222864958641, −7.111316938704445, −6.550926264288718, −6.212671926806157, −5.465471114911217, −4.730347664536197, −3.983110183524777, −3.473792039984687, −2.816739764707330, −1.741022622465732, −1.171921253410920, 0,
1.171921253410920, 1.741022622465732, 2.816739764707330, 3.473792039984687, 3.983110183524777, 4.730347664536197, 5.465471114911217, 6.212671926806157, 6.550926264288718, 7.111316938704445, 8.086222864958641, 8.372135528135169, 9.098122557296207, 9.541820128485428, 10.12455145454971, 10.97610983071725, 11.14755537879427, 11.85452918000077, 12.35412752101343, 13.09058458349636, 13.53640869231775, 13.85941037768347, 14.54772231600569, 15.13656378010121, 15.88541683638899