L(s) = 1 | + 4·5-s − 2·11-s + 6·13-s − 4·17-s + 4·19-s − 2·23-s + 11·25-s + 2·29-s + 2·37-s − 4·43-s + 12·47-s + 6·53-s − 8·55-s − 8·59-s − 6·61-s + 24·65-s − 8·67-s − 14·71-s + 2·73-s + 12·79-s − 4·83-s − 16·85-s + 16·95-s + 2·97-s + 16·101-s + 16·103-s − 18·107-s + ⋯ |
L(s) = 1 | + 1.78·5-s − 0.603·11-s + 1.66·13-s − 0.970·17-s + 0.917·19-s − 0.417·23-s + 11/5·25-s + 0.371·29-s + 0.328·37-s − 0.609·43-s + 1.75·47-s + 0.824·53-s − 1.07·55-s − 1.04·59-s − 0.768·61-s + 2.97·65-s − 0.977·67-s − 1.66·71-s + 0.234·73-s + 1.35·79-s − 0.439·83-s − 1.73·85-s + 1.64·95-s + 0.203·97-s + 1.59·101-s + 1.57·103-s − 1.74·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.546503206\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.546503206\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 4 T + p T^{2} \) |
| 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
−9.133591619397453241615790856366, −8.847117284114771599871123870971, −7.74475735714765512283633749961, −6.66397314528471112905733886770, −6.01041095937836104246896195058, −5.47944807329373236781153563032, −4.44067514856766940885014862243, −3.15949197761990841887540386198, −2.19665073610568722129910000758, −1.21434687808983120631223068182,
1.21434687808983120631223068182, 2.19665073610568722129910000758, 3.15949197761990841887540386198, 4.44067514856766940885014862243, 5.47944807329373236781153563032, 6.01041095937836104246896195058, 6.66397314528471112905733886770, 7.74475735714765512283633749961, 8.847117284114771599871123870971, 9.133591619397453241615790856366