| L(s) = 1 | − 7-s + 4·11-s + 6·13-s + 2·17-s + 6·19-s + 2·23-s − 6·29-s − 2·31-s + 4·37-s − 8·41-s + 4·43-s + 4·47-s + 49-s + 6·53-s − 4·59-s + 14·61-s − 4·67-s + 10·73-s − 4·77-s − 16·83-s − 8·89-s − 6·91-s − 10·97-s − 16·101-s − 16·103-s − 6·107-s − 14·109-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 1.20·11-s + 1.66·13-s + 0.485·17-s + 1.37·19-s + 0.417·23-s − 1.11·29-s − 0.359·31-s + 0.657·37-s − 1.24·41-s + 0.609·43-s + 0.583·47-s + 1/7·49-s + 0.824·53-s − 0.520·59-s + 1.79·61-s − 0.488·67-s + 1.17·73-s − 0.455·77-s − 1.75·83-s − 0.847·89-s − 0.628·91-s − 1.01·97-s − 1.59·101-s − 1.57·103-s − 0.580·107-s − 1.34·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.507190893\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.507190893\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.131890706453948184570671350528, −7.16210467109688053377700596946, −6.73729654778588428291582758126, −5.78786571439662084834813964485, −5.46505875227051947526810291933, −4.14934242766918376469827701972, −3.68069679041887771932599163856, −2.96269561686820682512049306286, −1.60945577397969897700417016725, −0.909971821433674117280248571734,
0.909971821433674117280248571734, 1.60945577397969897700417016725, 2.96269561686820682512049306286, 3.68069679041887771932599163856, 4.14934242766918376469827701972, 5.46505875227051947526810291933, 5.78786571439662084834813964485, 6.73729654778588428291582758126, 7.16210467109688053377700596946, 8.131890706453948184570671350528
