| 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
| \(L(2)\) |
\(\approx\) |
\(0.5531858757\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5531858757\) |
| \(L(\frac{5}{2})\) |
|
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
−8.831046824960305760015595849997, −8.160699968464707852836897219361, −7.30673831476543661885804844380, −6.93745290354349801068578500948, −5.91618449669187049781071790561, −4.82841801617000702709959491846, −3.61452694689995193397884259660, −2.29644527089062935172324900369, −1.69058239451996903867769637607, −0.44074670752905130658629967767,
0.44074670752905130658629967767, 1.69058239451996903867769637607, 2.29644527089062935172324900369, 3.61452694689995193397884259660, 4.82841801617000702709959491846, 5.91618449669187049781071790561, 6.93745290354349801068578500948, 7.30673831476543661885804844380, 8.160699968464707852836897219361, 8.831046824960305760015595849997
