| L(s) = 1 | − 1.02·3-s + 3.56·5-s + 7-s − 1.94·9-s − 11-s + 13-s − 3.66·15-s − 2.59·17-s + 1.78·19-s − 1.02·21-s − 2.86·23-s + 7.73·25-s + 5.07·27-s + 4.51·29-s + 6.44·31-s + 1.02·33-s + 3.56·35-s − 2.49·37-s − 1.02·39-s − 8.70·41-s − 1.37·43-s − 6.94·45-s + 7.82·47-s + 49-s + 2.66·51-s − 9.61·53-s − 3.56·55-s + ⋯ |
| L(s) = 1 | − 0.592·3-s + 1.59·5-s + 0.377·7-s − 0.649·9-s − 0.301·11-s + 0.277·13-s − 0.945·15-s − 0.630·17-s + 0.409·19-s − 0.223·21-s − 0.596·23-s + 1.54·25-s + 0.976·27-s + 0.838·29-s + 1.15·31-s + 0.178·33-s + 0.603·35-s − 0.409·37-s − 0.164·39-s − 1.36·41-s − 0.210·43-s − 1.03·45-s + 1.14·47-s + 0.142·49-s + 0.373·51-s − 1.32·53-s − 0.481·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.180689655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.180689655\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 1.02T + 3T^{2} \) |
| 5 | \( 1 - 3.56T + 5T^{2} \) |
| 17 | \( 1 + 2.59T + 17T^{2} \) |
| 19 | \( 1 - 1.78T + 19T^{2} \) |
| 23 | \( 1 + 2.86T + 23T^{2} \) |
| 29 | \( 1 - 4.51T + 29T^{2} \) |
| 31 | \( 1 - 6.44T + 31T^{2} \) |
| 37 | \( 1 + 2.49T + 37T^{2} \) |
| 41 | \( 1 + 8.70T + 41T^{2} \) |
| 43 | \( 1 + 1.37T + 43T^{2} \) |
| 47 | \( 1 - 7.82T + 47T^{2} \) |
| 53 | \( 1 + 9.61T + 53T^{2} \) |
| 59 | \( 1 + 2.27T + 59T^{2} \) |
| 61 | \( 1 + 3.88T + 61T^{2} \) |
| 67 | \( 1 - 15.1T + 67T^{2} \) |
| 71 | \( 1 - 15.3T + 71T^{2} \) |
| 73 | \( 1 + 5.51T + 73T^{2} \) |
| 79 | \( 1 + 2.13T + 79T^{2} \) |
| 83 | \( 1 - 11.8T + 83T^{2} \) |
| 89 | \( 1 - 6.46T + 89T^{2} \) |
| 97 | \( 1 + 7.82T + 97T^{2} \) |
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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
−7.963893447632809778354765409580, −6.80669807966878703499543114941, −6.35366114089701077106751374035, −5.79410367319443408761795060154, −5.12077068259319973355338060003, −4.66156052105723757523989233932, −3.36414668391180792682960439649, −2.49876280162509506036402762622, −1.82424467045937441458474244848, −0.75656204263865192676016690147,
0.75656204263865192676016690147, 1.82424467045937441458474244848, 2.49876280162509506036402762622, 3.36414668391180792682960439649, 4.66156052105723757523989233932, 5.12077068259319973355338060003, 5.79410367319443408761795060154, 6.35366114089701077106751374035, 6.80669807966878703499543114941, 7.963893447632809778354765409580
