| L(s) = 1 | + 2-s − 2.82·3-s − 4-s − 2.82·6-s − 3·8-s + 5.00·9-s + 2.82·12-s − 4.24·13-s − 16-s − 4.24·17-s + 5.00·18-s − 2.82·19-s + 4·23-s + 8.48·24-s − 4.24·26-s − 5.65·27-s + 5.65·31-s + 5·32-s − 4.24·34-s − 5.00·36-s + 6·37-s − 2.82·38-s + 12·39-s + 4.24·41-s + 4·46-s + 2.82·48-s + 12·51-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.63·3-s − 0.5·4-s − 1.15·6-s − 1.06·8-s + 1.66·9-s + 0.816·12-s − 1.17·13-s − 0.250·16-s − 1.02·17-s + 1.17·18-s − 0.648·19-s + 0.834·23-s + 1.73·24-s − 0.832·26-s − 1.08·27-s + 1.01·31-s + 0.883·32-s − 0.727·34-s − 0.833·36-s + 0.986·37-s − 0.458·38-s + 1.92·39-s + 0.662·41-s + 0.589·46-s + 0.408·48-s + 1.68·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7308832652\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7308832652\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - T + 2T^{2} \) |
| 3 | \( 1 + 2.82T + 3T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 + 4.24T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 - 4.24T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 8T + 53T^{2} \) |
| 59 | \( 1 - 8.48T + 59T^{2} \) |
| 61 | \( 1 - 9.89T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 - 8.48T + 83T^{2} \) |
| 89 | \( 1 + 4.24T + 89T^{2} \) |
| 97 | \( 1 - 4.24T + 97T^{2} \) |
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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
−9.837994967643485875386341524897, −9.120243484267251409314341815232, −7.969689087758694515812449954851, −6.75816224192841022856651307800, −6.30078858233938875674405760206, −5.26845740515687726986433825681, −4.79446893628447739625188208906, −4.06548061239794633931168717861, −2.53909939841841286474128470107, −0.60370243124921022818705555720,
0.60370243124921022818705555720, 2.53909939841841286474128470107, 4.06548061239794633931168717861, 4.79446893628447739625188208906, 5.26845740515687726986433825681, 6.30078858233938875674405760206, 6.75816224192841022856651307800, 7.969689087758694515812449954851, 9.120243484267251409314341815232, 9.837994967643485875386341524897
