| L(s) = 1 | + 3·3-s − 5·5-s − 8·7-s + 9·9-s + 20·11-s − 22·13-s − 15·15-s − 14·17-s + 76·19-s − 24·21-s − 56·23-s + 25·25-s + 27·27-s + 154·29-s − 160·31-s + 60·33-s + 40·35-s + 162·37-s − 66·39-s − 390·41-s + 388·43-s − 45·45-s + 544·47-s − 279·49-s − 42·51-s + 210·53-s − 100·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.431·7-s + 1/3·9-s + 0.548·11-s − 0.469·13-s − 0.258·15-s − 0.199·17-s + 0.917·19-s − 0.249·21-s − 0.507·23-s + 1/5·25-s + 0.192·27-s + 0.986·29-s − 0.926·31-s + 0.316·33-s + 0.193·35-s + 0.719·37-s − 0.270·39-s − 1.48·41-s + 1.37·43-s − 0.149·45-s + 1.68·47-s − 0.813·49-s − 0.115·51-s + 0.544·53-s − 0.245·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.206689934\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.206689934\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( 1 + 8 T + p^{3} T^{2} \) |
| 11 | \( 1 - 20 T + p^{3} T^{2} \) |
| 13 | \( 1 + 22 T + p^{3} T^{2} \) |
| 17 | \( 1 + 14 T + p^{3} T^{2} \) |
| 19 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 23 | \( 1 + 56 T + p^{3} T^{2} \) |
| 29 | \( 1 - 154 T + p^{3} T^{2} \) |
| 31 | \( 1 + 160 T + p^{3} T^{2} \) |
| 37 | \( 1 - 162 T + p^{3} T^{2} \) |
| 41 | \( 1 + 390 T + p^{3} T^{2} \) |
| 43 | \( 1 - 388 T + p^{3} T^{2} \) |
| 47 | \( 1 - 544 T + p^{3} T^{2} \) |
| 53 | \( 1 - 210 T + p^{3} T^{2} \) |
| 59 | \( 1 + 380 T + p^{3} T^{2} \) |
| 61 | \( 1 - 794 T + p^{3} T^{2} \) |
| 67 | \( 1 + 148 T + p^{3} T^{2} \) |
| 71 | \( 1 - 840 T + p^{3} T^{2} \) |
| 73 | \( 1 - 858 T + p^{3} T^{2} \) |
| 79 | \( 1 + 144 T + p^{3} T^{2} \) |
| 83 | \( 1 - 316 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1098 T + p^{3} T^{2} \) |
| 97 | \( 1 - 994 T + p^{3} T^{2} \) |
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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
−9.544189485636355639647850763945, −8.881028320719371194949530883505, −7.942896520420428631160856150641, −7.21515153342031165469286705907, −6.37618378309838914063990451506, −5.20263630108583489030863330740, −4.12132872074058698147333633908, −3.31970262328451686202045109336, −2.23100550544409738953730958002, −0.77757767170116201164117584102,
0.77757767170116201164117584102, 2.23100550544409738953730958002, 3.31970262328451686202045109336, 4.12132872074058698147333633908, 5.20263630108583489030863330740, 6.37618378309838914063990451506, 7.21515153342031165469286705907, 7.942896520420428631160856150641, 8.881028320719371194949530883505, 9.544189485636355639647850763945
