L(s) = 1 | + 5.02·2-s + 17.2·4-s − 5.38·7-s + 46.4·8-s + 39.1·11-s + 86.6·13-s − 27.0·14-s + 95.2·16-s − 15.4·17-s − 26.8·19-s + 196.·22-s + 111.·23-s + 435.·26-s − 92.7·28-s − 49.2·29-s − 179.·31-s + 107.·32-s − 77.4·34-s + 293.·37-s − 134.·38-s − 27.6·41-s + 60.5·43-s + 674.·44-s + 558.·46-s − 96.2·47-s − 314.·49-s + 1.49e3·52-s + ⋯ |
L(s) = 1 | + 1.77·2-s + 2.15·4-s − 0.290·7-s + 2.05·8-s + 1.07·11-s + 1.84·13-s − 0.516·14-s + 1.48·16-s − 0.219·17-s − 0.324·19-s + 1.90·22-s + 1.00·23-s + 3.28·26-s − 0.626·28-s − 0.315·29-s − 1.04·31-s + 0.592·32-s − 0.390·34-s + 1.30·37-s − 0.575·38-s − 0.105·41-s + 0.214·43-s + 2.30·44-s + 1.78·46-s − 0.298·47-s − 0.915·49-s + 3.98·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(8.425258931\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.425258931\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 5.02T + 8T^{2} \) |
| 7 | \( 1 + 5.38T + 343T^{2} \) |
| 11 | \( 1 - 39.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 86.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 15.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 26.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 111.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 49.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 179.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 293.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 27.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 60.5T + 7.95e4T^{2} \) |
| 47 | \( 1 + 96.2T + 1.03e5T^{2} \) |
| 53 | \( 1 + 251.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 76.8T + 2.05e5T^{2} \) |
| 61 | \( 1 - 490.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 238.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 640.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 769.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 662.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.30e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 995.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 814.T + 9.12e5T^{2} \) |
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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
−8.806826272931977436241035926126, −7.75853220581087934856503512480, −6.59688322113967819023755503743, −6.43111125776631471707259885130, −5.56167480013210308125083071983, −4.65155942790625785454976534812, −3.73783396997568039229950301669, −3.41655154248184077556449318750, −2.14745254780300964819059754413, −1.09605918350408668000802006064,
1.09605918350408668000802006064, 2.14745254780300964819059754413, 3.41655154248184077556449318750, 3.73783396997568039229950301669, 4.65155942790625785454976534812, 5.56167480013210308125083071983, 6.43111125776631471707259885130, 6.59688322113967819023755503743, 7.75853220581087934856503512480, 8.806826272931977436241035926126