L(s) = 1 | − 2-s + 2.65·3-s + 4-s − 2.81·5-s − 2.65·6-s − 1.05·7-s − 8-s + 4.05·9-s + 2.81·10-s − 3.80·11-s + 2.65·12-s − 2.54·13-s + 1.05·14-s − 7.48·15-s + 16-s + 3.72·17-s − 4.05·18-s − 19-s − 2.81·20-s − 2.79·21-s + 3.80·22-s − 1.06·23-s − 2.65·24-s + 2.94·25-s + 2.54·26-s + 2.81·27-s − 1.05·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.53·3-s + 0.5·4-s − 1.26·5-s − 1.08·6-s − 0.397·7-s − 0.353·8-s + 1.35·9-s + 0.891·10-s − 1.14·11-s + 0.766·12-s − 0.704·13-s + 0.280·14-s − 1.93·15-s + 0.250·16-s + 0.902·17-s − 0.956·18-s − 0.229·19-s − 0.630·20-s − 0.609·21-s + 0.810·22-s − 0.221·23-s − 0.542·24-s + 0.589·25-s + 0.498·26-s + 0.541·27-s − 0.198·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.299110320\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.299110320\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 - T \) |
good | 3 | \( 1 - 2.65T + 3T^{2} \) |
| 5 | \( 1 + 2.81T + 5T^{2} \) |
| 7 | \( 1 + 1.05T + 7T^{2} \) |
| 11 | \( 1 + 3.80T + 11T^{2} \) |
| 13 | \( 1 + 2.54T + 13T^{2} \) |
| 17 | \( 1 - 3.72T + 17T^{2} \) |
| 23 | \( 1 + 1.06T + 23T^{2} \) |
| 29 | \( 1 - 6.91T + 29T^{2} \) |
| 31 | \( 1 + 2.89T + 31T^{2} \) |
| 37 | \( 1 + 4.68T + 37T^{2} \) |
| 41 | \( 1 + 7.51T + 41T^{2} \) |
| 43 | \( 1 - 5.56T + 43T^{2} \) |
| 47 | \( 1 - 6.82T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 + 6.26T + 59T^{2} \) |
| 61 | \( 1 - 8.07T + 61T^{2} \) |
| 67 | \( 1 - 3.37T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 - 4.75T + 73T^{2} \) |
| 79 | \( 1 - 4.31T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 - 9.38T + 89T^{2} \) |
| 97 | \( 1 - 3.46T + 97T^{2} \) |
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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
−8.041941671439444238374880384253, −7.30358153543805119471191408262, −7.09121061544780714512594946286, −5.82214878652237089879241164369, −4.87152144452205847080034396080, −3.97373154716846655816777671520, −3.27851535767294191610006720240, −2.76177202268378503853542367761, −1.95620259330256572605770146333, −0.55274977086561654102753961811,
0.55274977086561654102753961811, 1.95620259330256572605770146333, 2.76177202268378503853542367761, 3.27851535767294191610006720240, 3.97373154716846655816777671520, 4.87152144452205847080034396080, 5.82214878652237089879241164369, 7.09121061544780714512594946286, 7.30358153543805119471191408262, 8.041941671439444238374880384253