| L(s) = 1 | − 4.90·2-s + 16.1·4-s + 2.10·7-s − 39.7·8-s + 40.3·11-s + 61.7·13-s − 10.3·14-s + 66.5·16-s − 99.2·17-s + 134.·19-s − 197.·22-s − 76.4·23-s − 303.·26-s + 33.8·28-s + 236.·29-s + 243.·31-s − 8.27·32-s + 487.·34-s + 57.4·37-s − 660.·38-s − 411.·41-s + 102.·43-s + 649.·44-s + 375.·46-s + 260.·47-s − 338.·49-s + 994.·52-s + ⋯ |
| L(s) = 1 | − 1.73·2-s + 2.01·4-s + 0.113·7-s − 1.75·8-s + 1.10·11-s + 1.31·13-s − 0.197·14-s + 1.03·16-s − 1.41·17-s + 1.62·19-s − 1.91·22-s − 0.693·23-s − 2.28·26-s + 0.228·28-s + 1.51·29-s + 1.41·31-s − 0.0457·32-s + 2.45·34-s + 0.255·37-s − 2.81·38-s − 1.56·41-s + 0.364·43-s + 2.22·44-s + 1.20·46-s + 0.808·47-s − 0.987·49-s + 2.65·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.011951163\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.011951163\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 4.90T + 8T^{2} \) |
| 7 | \( 1 - 2.10T + 343T^{2} \) |
| 11 | \( 1 - 40.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 61.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 99.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 134.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 76.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 236.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 243.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 57.4T + 5.06e4T^{2} \) |
| 41 | \( 1 + 411.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 102.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 260.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 75.4T + 1.48e5T^{2} \) |
| 59 | \( 1 + 39.2T + 2.05e5T^{2} \) |
| 61 | \( 1 + 675.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 601.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 222.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 297.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 95.6T + 4.93e5T^{2} \) |
| 83 | \( 1 - 3.08T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.34e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.48e3T + 9.12e5T^{2} \) |
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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
−9.966209534330282307583862733229, −9.094948017171529039436944893565, −8.586915611400940022711131782860, −7.76454627256167357763963981239, −6.68013598294086809194357647346, −6.20036895870315709240743655571, −4.49449715127709360659026504732, −3.10380614250885153935170072674, −1.68312266626246378905505047641, −0.810388981678696396101618511468,
0.810388981678696396101618511468, 1.68312266626246378905505047641, 3.10380614250885153935170072674, 4.49449715127709360659026504732, 6.20036895870315709240743655571, 6.68013598294086809194357647346, 7.76454627256167357763963981239, 8.586915611400940022711131782860, 9.094948017171529039436944893565, 9.966209534330282307583862733229
