L(s) = 1 | − 3-s + 9-s + 4·11-s + 4·17-s + 4·23-s − 27-s − 6·29-s + 4·31-s − 4·33-s + 8·37-s + 10·41-s + 4·43-s + 4·47-s − 4·51-s + 12·53-s + 4·59-s − 2·61-s − 4·67-s − 4·69-s − 8·73-s + 12·79-s + 81-s − 4·83-s + 6·87-s + 10·89-s − 4·93-s + 8·97-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.20·11-s + 0.970·17-s + 0.834·23-s − 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.696·33-s + 1.31·37-s + 1.56·41-s + 0.609·43-s + 0.583·47-s − 0.560·51-s + 1.64·53-s + 0.520·59-s − 0.256·61-s − 0.488·67-s − 0.481·69-s − 0.936·73-s + 1.35·79-s + 1/9·81-s − 0.439·83-s + 0.643·87-s + 1.05·89-s − 0.414·93-s + 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.877788962\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.877788962\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−14.53788131159741, −13.83316553619959, −13.21186144909678, −12.83278794497122, −12.19271314816520, −11.77144230505563, −11.41907330107468, −10.74458874743655, −10.39293985542945, −9.511070262920879, −9.398148566253116, −8.750596155639762, −8.035310932997776, −7.358581642088123, −7.132503883799241, −6.234735937605448, −5.956610070709632, −5.386650597287903, −4.621647051761804, −4.112334013152602, −3.568087720641872, −2.781842648206040, −2.027038473288837, −1.068387894135325, −0.7505714890959849,
0.7505714890959849, 1.068387894135325, 2.027038473288837, 2.781842648206040, 3.568087720641872, 4.112334013152602, 4.621647051761804, 5.386650597287903, 5.956610070709632, 6.234735937605448, 7.132503883799241, 7.358581642088123, 8.035310932997776, 8.750596155639762, 9.398148566253116, 9.511070262920879, 10.39293985542945, 10.74458874743655, 11.41907330107468, 11.77144230505563, 12.19271314816520, 12.83278794497122, 13.21186144909678, 13.83316553619959, 14.53788131159741