L(s) = 1 | + 1.59·3-s − 2.83·5-s − 0.468·9-s + 5.68·11-s − 5.47·13-s − 4.51·15-s + 0.927·17-s + 7.70·19-s − 23-s + 3.03·25-s − 5.51·27-s − 0.155·29-s + 6.19·31-s + 9.04·33-s − 4.13·37-s − 8.70·39-s + 5.77·41-s − 8.58·43-s + 1.32·45-s + 1.52·47-s + 1.47·51-s − 3.51·53-s − 16.1·55-s + 12.2·57-s − 0.0446·59-s + 9.01·61-s + 15.5·65-s + ⋯ |
L(s) = 1 | + 0.918·3-s − 1.26·5-s − 0.156·9-s + 1.71·11-s − 1.51·13-s − 1.16·15-s + 0.224·17-s + 1.76·19-s − 0.208·23-s + 0.607·25-s − 1.06·27-s − 0.0289·29-s + 1.11·31-s + 1.57·33-s − 0.679·37-s − 1.39·39-s + 0.901·41-s − 1.30·43-s + 0.198·45-s + 0.223·47-s + 0.206·51-s − 0.482·53-s − 2.17·55-s + 1.62·57-s − 0.00581·59-s + 1.15·61-s + 1.92·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.075091777\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.075091777\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 - 1.59T + 3T^{2} \) |
| 5 | \( 1 + 2.83T + 5T^{2} \) |
| 11 | \( 1 - 5.68T + 11T^{2} \) |
| 13 | \( 1 + 5.47T + 13T^{2} \) |
| 17 | \( 1 - 0.927T + 17T^{2} \) |
| 19 | \( 1 - 7.70T + 19T^{2} \) |
| 29 | \( 1 + 0.155T + 29T^{2} \) |
| 31 | \( 1 - 6.19T + 31T^{2} \) |
| 37 | \( 1 + 4.13T + 37T^{2} \) |
| 41 | \( 1 - 5.77T + 41T^{2} \) |
| 43 | \( 1 + 8.58T + 43T^{2} \) |
| 47 | \( 1 - 1.52T + 47T^{2} \) |
| 53 | \( 1 + 3.51T + 53T^{2} \) |
| 59 | \( 1 + 0.0446T + 59T^{2} \) |
| 61 | \( 1 - 9.01T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 + 1.38T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 + 9.51T + 79T^{2} \) |
| 83 | \( 1 - 8.29T + 83T^{2} \) |
| 89 | \( 1 + 9.13T + 89T^{2} \) |
| 97 | \( 1 - 3.16T + 97T^{2} \) |
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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
−7.69338400447086484856457028856, −7.29978950600936412708312741121, −6.62364161635475499938807977419, −5.61727273211458751282402043502, −4.75542809287936137080535896694, −4.07789627823328024277963534839, −3.38151656555778265373251942119, −2.91069814727090002581537331012, −1.80173559123393126825214868461, −0.66460103034884772501532347371,
0.66460103034884772501532347371, 1.80173559123393126825214868461, 2.91069814727090002581537331012, 3.38151656555778265373251942119, 4.07789627823328024277963534839, 4.75542809287936137080535896694, 5.61727273211458751282402043502, 6.62364161635475499938807977419, 7.29978950600936412708312741121, 7.69338400447086484856457028856