| L(s) = 1 | + 3.14·3-s + 1.28·7-s + 6.91·9-s − 2.65·11-s − 5.53·13-s + 5.89·17-s + 6.95·19-s + 4.06·21-s + 1.47·23-s + 12.3·27-s + 4.21·29-s − 1.34·31-s − 8.36·33-s − 2.48·37-s − 17.4·39-s − 12.0·41-s + 5.14·43-s − 3.04·47-s − 5.33·49-s + 18.5·51-s + 10.4·53-s + 21.8·57-s − 5.12·59-s − 8.21·61-s + 8.92·63-s − 2.35·67-s + 4.64·69-s + ⋯ |
| L(s) = 1 | + 1.81·3-s + 0.487·7-s + 2.30·9-s − 0.800·11-s − 1.53·13-s + 1.42·17-s + 1.59·19-s + 0.886·21-s + 0.307·23-s + 2.37·27-s + 0.782·29-s − 0.241·31-s − 1.45·33-s − 0.408·37-s − 2.79·39-s − 1.88·41-s + 0.785·43-s − 0.443·47-s − 0.762·49-s + 2.59·51-s + 1.43·53-s + 2.90·57-s − 0.667·59-s − 1.05·61-s + 1.12·63-s − 0.288·67-s + 0.558·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.077867382\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.077867382\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 3.14T + 3T^{2} \) |
| 7 | \( 1 - 1.28T + 7T^{2} \) |
| 11 | \( 1 + 2.65T + 11T^{2} \) |
| 13 | \( 1 + 5.53T + 13T^{2} \) |
| 17 | \( 1 - 5.89T + 17T^{2} \) |
| 19 | \( 1 - 6.95T + 19T^{2} \) |
| 23 | \( 1 - 1.47T + 23T^{2} \) |
| 29 | \( 1 - 4.21T + 29T^{2} \) |
| 31 | \( 1 + 1.34T + 31T^{2} \) |
| 37 | \( 1 + 2.48T + 37T^{2} \) |
| 41 | \( 1 + 12.0T + 41T^{2} \) |
| 43 | \( 1 - 5.14T + 43T^{2} \) |
| 47 | \( 1 + 3.04T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + 5.12T + 59T^{2} \) |
| 61 | \( 1 + 8.21T + 61T^{2} \) |
| 67 | \( 1 + 2.35T + 67T^{2} \) |
| 71 | \( 1 + 8.48T + 71T^{2} \) |
| 73 | \( 1 + 5.88T + 73T^{2} \) |
| 79 | \( 1 + 16.2T + 79T^{2} \) |
| 83 | \( 1 + 0.931T + 83T^{2} \) |
| 89 | \( 1 + 0.505T + 89T^{2} \) |
| 97 | \( 1 - 8.37T + 97T^{2} \) |
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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
−9.963179498653244629830346419972, −9.102275369853128783113347983356, −8.232436366801331802005855406671, −7.56050762540458829890118460434, −7.19265716290450324914219984048, −5.39263530733613974597267179770, −4.63617267667355667150949487296, −3.28397634193518396177328472570, −2.76438507802768423978776991107, −1.53416430946282725536633804121,
1.53416430946282725536633804121, 2.76438507802768423978776991107, 3.28397634193518396177328472570, 4.63617267667355667150949487296, 5.39263530733613974597267179770, 7.19265716290450324914219984048, 7.56050762540458829890118460434, 8.232436366801331802005855406671, 9.102275369853128783113347983356, 9.963179498653244629830346419972
