| L(s) = 1 | + 4·2-s + 16·4-s + 4·7-s + 64·8-s + 500·11-s + 288·13-s + 16·14-s + 256·16-s + 1.51e3·17-s − 1.34e3·19-s + 2.00e3·22-s − 4.10e3·23-s + 1.15e3·26-s + 64·28-s + 2.64e3·29-s − 5.61e3·31-s + 1.02e3·32-s + 6.06e3·34-s + 7.28e3·37-s − 5.37e3·38-s + 1.89e4·41-s + 2.40e3·43-s + 8.00e3·44-s − 1.64e4·46-s + 8.90e3·47-s − 1.67e4·49-s + 4.60e3·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.0308·7-s + 0.353·8-s + 1.24·11-s + 0.472·13-s + 0.0218·14-s + 1/4·16-s + 1.27·17-s − 0.854·19-s + 0.880·22-s − 1.61·23-s + 0.334·26-s + 0.0154·28-s + 0.584·29-s − 1.04·31-s + 0.176·32-s + 0.899·34-s + 0.875·37-s − 0.603·38-s + 1.76·41-s + 0.198·43-s + 0.622·44-s − 1.14·46-s + 0.587·47-s − 0.999·49-s + 0.236·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.977835754\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.977835754\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p^{5} T^{2} \) |
| 11 | \( 1 - 500 T + p^{5} T^{2} \) |
| 13 | \( 1 - 288 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1516 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1344 T + p^{5} T^{2} \) |
| 23 | \( 1 + 4100 T + p^{5} T^{2} \) |
| 29 | \( 1 - 2646 T + p^{5} T^{2} \) |
| 31 | \( 1 + 5612 T + p^{5} T^{2} \) |
| 37 | \( 1 - 7288 T + p^{5} T^{2} \) |
| 41 | \( 1 - 18986 T + p^{5} T^{2} \) |
| 43 | \( 1 - 2404 T + p^{5} T^{2} \) |
| 47 | \( 1 - 8900 T + p^{5} T^{2} \) |
| 53 | \( 1 - 39804 T + p^{5} T^{2} \) |
| 59 | \( 1 - 28300 T + p^{5} T^{2} \) |
| 61 | \( 1 - 18290 T + p^{5} T^{2} \) |
| 67 | \( 1 + 65956 T + p^{5} T^{2} \) |
| 71 | \( 1 - 28800 T + p^{5} T^{2} \) |
| 73 | \( 1 - 30808 T + p^{5} T^{2} \) |
| 79 | \( 1 - 60228 T + p^{5} T^{2} \) |
| 83 | \( 1 + 2468 T + p^{5} T^{2} \) |
| 89 | \( 1 + 22678 T + p^{5} T^{2} \) |
| 97 | \( 1 - 36968 T + p^{5} T^{2} \) |
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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
−10.39264433003353063776931995034, −9.463892403269184958563768932466, −8.392706017269208321721586978162, −7.41723788460385208448537914470, −6.30937233042952757078267505617, −5.68801266703891578417915951527, −4.27104047231313268188245444439, −3.63773101676021423913711963197, −2.20112867875869827023421700137, −0.974202007876161717246547218173,
0.974202007876161717246547218173, 2.20112867875869827023421700137, 3.63773101676021423913711963197, 4.27104047231313268188245444439, 5.68801266703891578417915951527, 6.30937233042952757078267505617, 7.41723788460385208448537914470, 8.392706017269208321721586978162, 9.463892403269184958563768932466, 10.39264433003353063776931995034
