| L(s) = 1 | + 2.02·2-s − 3-s + 2.09·4-s + 5-s − 2.02·6-s + 0.186·8-s + 9-s + 2.02·10-s + 11-s − 2.09·12-s − 0.386·13-s − 15-s − 3.80·16-s − 4.09·17-s + 2.02·18-s + 8.07·19-s + 2.09·20-s + 2.02·22-s + 4.54·23-s − 0.186·24-s + 25-s − 0.781·26-s − 27-s + 9.37·29-s − 2.02·30-s − 3.88·31-s − 8.07·32-s + ⋯ |
| L(s) = 1 | + 1.43·2-s − 0.577·3-s + 1.04·4-s + 0.447·5-s − 0.825·6-s + 0.0658·8-s + 0.333·9-s + 0.639·10-s + 0.301·11-s − 0.603·12-s − 0.107·13-s − 0.258·15-s − 0.951·16-s − 0.993·17-s + 0.476·18-s + 1.85·19-s + 0.467·20-s + 0.431·22-s + 0.947·23-s − 0.0380·24-s + 0.200·25-s − 0.153·26-s − 0.192·27-s + 1.74·29-s − 0.369·30-s − 0.697·31-s − 1.42·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.097077483\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.097077483\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 2.02T + 2T^{2} \) |
| 13 | \( 1 + 0.386T + 13T^{2} \) |
| 17 | \( 1 + 4.09T + 17T^{2} \) |
| 19 | \( 1 - 8.07T + 19T^{2} \) |
| 23 | \( 1 - 4.54T + 23T^{2} \) |
| 29 | \( 1 - 9.37T + 29T^{2} \) |
| 31 | \( 1 + 3.88T + 31T^{2} \) |
| 37 | \( 1 - 1.42T + 37T^{2} \) |
| 41 | \( 1 + 4.61T + 41T^{2} \) |
| 43 | \( 1 - 0.660T + 43T^{2} \) |
| 47 | \( 1 + 6.42T + 47T^{2} \) |
| 53 | \( 1 - 6.09T + 53T^{2} \) |
| 59 | \( 1 - 0.213T + 59T^{2} \) |
| 61 | \( 1 - 7.06T + 61T^{2} \) |
| 67 | \( 1 + 8.30T + 67T^{2} \) |
| 71 | \( 1 - 3.42T + 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 + 8.62T + 83T^{2} \) |
| 89 | \( 1 - 8.66T + 89T^{2} \) |
| 97 | \( 1 - 7.90T + 97T^{2} \) |
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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
−7.37744150107184784932628427995, −6.88203683993947850955125941138, −6.24283940757946770785481114073, −5.63213893846803232556015269784, −4.89168424913588391700841252900, −4.63035264303904932852976968514, −3.54024246467105074189969541240, −2.96864593057597791695935189427, −1.99900217880705796447348129527, −0.837007235940796303957006202119,
0.837007235940796303957006202119, 1.99900217880705796447348129527, 2.96864593057597791695935189427, 3.54024246467105074189969541240, 4.63035264303904932852976968514, 4.89168424913588391700841252900, 5.63213893846803232556015269784, 6.24283940757946770785481114073, 6.88203683993947850955125941138, 7.37744150107184784932628427995
