| L(s) = 1 | − 5·11-s − 3·13-s + 17-s + 6·19-s − 6·23-s + 9·29-s − 4·31-s − 2·37-s − 4·41-s + 10·43-s − 47-s + 4·53-s + 8·59-s + 8·61-s + 12·67-s + 8·71-s + 2·73-s − 13·79-s − 4·83-s + 4·89-s − 13·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.50·11-s − 0.832·13-s + 0.242·17-s + 1.37·19-s − 1.25·23-s + 1.67·29-s − 0.718·31-s − 0.328·37-s − 0.624·41-s + 1.52·43-s − 0.145·47-s + 0.549·53-s + 1.04·59-s + 1.02·61-s + 1.46·67-s + 0.949·71-s + 0.234·73-s − 1.46·79-s − 0.439·83-s + 0.423·89-s − 1.31·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.680488842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.680488842\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−13.16891500121363, −12.54306066372553, −12.40477027731988, −11.78107825948363, −11.35171362989300, −10.77412714869807, −10.19323386339964, −9.920050304934448, −9.621674517901490, −8.773204336931352, −8.338417789674164, −7.857283135028049, −7.455317566326755, −6.989440623436744, −6.404242437490152, −5.593482703291850, −5.366464221299756, −4.950973683340069, −4.210425914924096, −3.679728446746072, −2.930883034409772, −2.540270788580640, −2.014668264349887, −1.072365162183534, −0.4066618816742264,
0.4066618816742264, 1.072365162183534, 2.014668264349887, 2.540270788580640, 2.930883034409772, 3.679728446746072, 4.210425914924096, 4.950973683340069, 5.366464221299756, 5.593482703291850, 6.404242437490152, 6.989440623436744, 7.455317566326755, 7.857283135028049, 8.338417789674164, 8.773204336931352, 9.621674517901490, 9.920050304934448, 10.19323386339964, 10.77412714869807, 11.35171362989300, 11.78107825948363, 12.40477027731988, 12.54306066372553, 13.16891500121363
