| L(s) = 1 | − 3.56·2-s + 4.68·4-s + 6.06·7-s + 11.8·8-s + 61.3·11-s + 70.8·13-s − 21.5·14-s − 79.5·16-s + 51.5·17-s + 15.7·19-s − 218.·22-s + 114.·23-s − 252.·26-s + 28.4·28-s + 39.7·29-s − 240.·31-s + 188.·32-s − 183.·34-s + 129.·37-s − 56.2·38-s + 262.·41-s − 127.·43-s + 287.·44-s − 409.·46-s + 85.2·47-s − 306.·49-s + 331.·52-s + ⋯ |
| L(s) = 1 | − 1.25·2-s + 0.585·4-s + 0.327·7-s + 0.521·8-s + 1.68·11-s + 1.51·13-s − 0.412·14-s − 1.24·16-s + 0.735·17-s + 0.190·19-s − 2.11·22-s + 1.04·23-s − 1.90·26-s + 0.191·28-s + 0.254·29-s − 1.39·31-s + 1.04·32-s − 0.926·34-s + 0.575·37-s − 0.240·38-s + 0.999·41-s − 0.452·43-s + 0.985·44-s − 1.31·46-s + 0.264·47-s − 0.892·49-s + 0.885·52-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
| \(L(2)\) |
\(\approx\) |
\(1.359405773\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.359405773\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 3.56T + 8T^{2} \) |
| 7 | \( 1 - 6.06T + 343T^{2} \) |
| 11 | \( 1 - 61.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 70.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 51.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 15.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 114.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 39.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 240.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 129.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 262.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 127.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 85.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 328.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 687.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 455.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 955.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 160.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 702.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 65.3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 996.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.33e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.40e3T + 9.12e5T^{2} \) |
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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
−9.860175833184925039615350136480, −9.101294795320679133815078691352, −8.642334263939305214474846228761, −7.67036876205642156183495845006, −6.80861553704977853034480077160, −5.84334107776949058250656038836, −4.44679920774628318142704041406, −3.44339984376492101779521972598, −1.59590803678309816881028195627, −0.937882546359165829616370828242,
0.937882546359165829616370828242, 1.59590803678309816881028195627, 3.44339984376492101779521972598, 4.44679920774628318142704041406, 5.84334107776949058250656038836, 6.80861553704977853034480077160, 7.67036876205642156183495845006, 8.642334263939305214474846228761, 9.101294795320679133815078691352, 9.860175833184925039615350136480
