| L(s) = 1 | + 3·3-s + 11·5-s + 9·9-s + 39·11-s + 32·13-s + 33·15-s − 12·17-s + 88·19-s − 92·23-s − 4·25-s + 27·27-s + 255·29-s + 35·31-s + 117·33-s − 4·37-s + 96·39-s − 16·41-s − 330·43-s + 99·45-s + 298·47-s − 36·51-s − 717·53-s + 429·55-s + 264·57-s + 217·59-s − 386·61-s + 352·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.983·5-s + 1/3·9-s + 1.06·11-s + 0.682·13-s + 0.568·15-s − 0.171·17-s + 1.06·19-s − 0.834·23-s − 0.0319·25-s + 0.192·27-s + 1.63·29-s + 0.202·31-s + 0.617·33-s − 0.0177·37-s + 0.394·39-s − 0.0609·41-s − 1.17·43-s + 0.327·45-s + 0.924·47-s − 0.0988·51-s − 1.85·53-s + 1.05·55-s + 0.613·57-s + 0.478·59-s − 0.810·61-s + 0.671·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.853816786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.853816786\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 11 T + p^{3} T^{2} \) |
| 11 | \( 1 - 39 T + p^{3} T^{2} \) |
| 13 | \( 1 - 32 T + p^{3} T^{2} \) |
| 17 | \( 1 + 12 T + p^{3} T^{2} \) |
| 19 | \( 1 - 88 T + p^{3} T^{2} \) |
| 23 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 29 | \( 1 - 255 T + p^{3} T^{2} \) |
| 31 | \( 1 - 35 T + p^{3} T^{2} \) |
| 37 | \( 1 + 4 T + p^{3} T^{2} \) |
| 41 | \( 1 + 16 T + p^{3} T^{2} \) |
| 43 | \( 1 + 330 T + p^{3} T^{2} \) |
| 47 | \( 1 - 298 T + p^{3} T^{2} \) |
| 53 | \( 1 + 717 T + p^{3} T^{2} \) |
| 59 | \( 1 - 217 T + p^{3} T^{2} \) |
| 61 | \( 1 + 386 T + p^{3} T^{2} \) |
| 67 | \( 1 - 906 T + p^{3} T^{2} \) |
| 71 | \( 1 + 34 T + p^{3} T^{2} \) |
| 73 | \( 1 - 838 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1325 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1163 T + p^{3} T^{2} \) |
| 89 | \( 1 - 54 T + p^{3} T^{2} \) |
| 97 | \( 1 + 7 T + p^{3} T^{2} \) |
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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
−9.525982551796681618088497572617, −8.640010848924247323792979736005, −7.921934371850615589033788421856, −6.72078134701982355303624098698, −6.20869164320751776850176568365, −5.16059878152039148657429400540, −4.05745900595418240399282190905, −3.12060064012223091370606529921, −1.96745920067686375638883724048, −1.07279554813215444333418936590,
1.07279554813215444333418936590, 1.96745920067686375638883724048, 3.12060064012223091370606529921, 4.05745900595418240399282190905, 5.16059878152039148657429400540, 6.20869164320751776850176568365, 6.72078134701982355303624098698, 7.921934371850615589033788421856, 8.640010848924247323792979736005, 9.525982551796681618088497572617
