| L(s) = 1 | + 1.56·2-s − 5.56·4-s + 24.2·7-s − 21.1·8-s + 11·11-s + 84.2·13-s + 37.8·14-s + 11.4·16-s + 40.9·17-s + 120.·19-s + 17.1·22-s + 9.94·23-s + 131.·26-s − 134.·28-s − 196.·29-s + 151.·31-s + 187.·32-s + 63.9·34-s − 253.·37-s + 188.·38-s + 179.·41-s − 90.4·43-s − 61.1·44-s + 15.5·46-s + 483.·47-s + 245.·49-s − 468.·52-s + ⋯ |
| L(s) = 1 | + 0.552·2-s − 0.695·4-s + 1.31·7-s − 0.935·8-s + 0.301·11-s + 1.79·13-s + 0.723·14-s + 0.178·16-s + 0.584·17-s + 1.45·19-s + 0.166·22-s + 0.0901·23-s + 0.992·26-s − 0.910·28-s − 1.26·29-s + 0.875·31-s + 1.03·32-s + 0.322·34-s − 1.12·37-s + 0.805·38-s + 0.683·41-s − 0.320·43-s − 0.209·44-s + 0.0497·46-s + 1.50·47-s + 0.716·49-s − 1.24·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.602869145\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.602869145\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 1.56T + 8T^{2} \) |
| 7 | \( 1 - 24.2T + 343T^{2} \) |
| 13 | \( 1 - 84.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 40.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 120.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 9.94T + 1.21e4T^{2} \) |
| 29 | \( 1 + 196.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 151.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 253.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 179.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 90.4T + 7.95e4T^{2} \) |
| 47 | \( 1 - 483.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 567.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 491.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 127.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 628.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 309.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.14e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 87.4T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.04e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 390.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 165.T + 9.12e5T^{2} \) |
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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
−8.539703249433004383025091194767, −7.980598778352472442557169671759, −7.07366354660709754990650320728, −5.83524269859890033442914012191, −5.51343465476860244164874620229, −4.58253718558698904786675485275, −3.83466365936903453268888405604, −3.12359792615209602630099063474, −1.58535481268965662317127937340, −0.862883092004760457964225198231,
0.862883092004760457964225198231, 1.58535481268965662317127937340, 3.12359792615209602630099063474, 3.83466365936903453268888405604, 4.58253718558698904786675485275, 5.51343465476860244164874620229, 5.83524269859890033442914012191, 7.07366354660709754990650320728, 7.980598778352472442557169671759, 8.539703249433004383025091194767
