| L(s) = 1 | + 3-s + 9-s + 11-s − 4·13-s + 6·17-s − 4·19-s + 6·23-s − 5·25-s + 27-s − 6·29-s − 8·31-s + 33-s + 10·37-s − 4·39-s − 6·41-s − 8·43-s + 6·47-s + 6·51-s − 4·57-s + 8·61-s + 4·67-s + 6·69-s + 6·71-s − 2·73-s − 5·75-s + 14·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.301·11-s − 1.10·13-s + 1.45·17-s − 0.917·19-s + 1.25·23-s − 25-s + 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.174·33-s + 1.64·37-s − 0.640·39-s − 0.937·41-s − 1.21·43-s + 0.875·47-s + 0.840·51-s − 0.529·57-s + 1.02·61-s + 0.488·67-s + 0.722·69-s + 0.712·71-s − 0.234·73-s − 0.577·75-s + 1.57·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.507210066\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.507210066\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.80178474185762, −13.12146540395034, −12.78147106983752, −12.42739274805430, −11.73566870927841, −11.32688210435597, −10.80307017440866, −10.05150738662099, −9.792743227854839, −9.344196382752240, −8.769409814270528, −8.254433537544419, −7.591468912877096, −7.383246222230420, −6.795067494094719, −6.099103055183169, −5.491011832507313, −5.046399583168376, −4.397255548015523, −3.629725918864200, −3.434888906374553, −2.516428948461398, −2.071120265796813, −1.360866126229138, −0.4731487391469865,
0.4731487391469865, 1.360866126229138, 2.071120265796813, 2.516428948461398, 3.434888906374553, 3.629725918864200, 4.397255548015523, 5.046399583168376, 5.491011832507313, 6.099103055183169, 6.795067494094719, 7.383246222230420, 7.591468912877096, 8.254433537544419, 8.769409814270528, 9.344196382752240, 9.792743227854839, 10.05150738662099, 10.80307017440866, 11.32688210435597, 11.73566870927841, 12.42739274805430, 12.78147106983752, 13.12146540395034, 13.80178474185762
