| L(s) = 1 | + 2.95·3-s + 5-s − 3.95·7-s + 5.74·9-s + 0.957·11-s − 2.74·13-s + 2.95·15-s + 5.74·17-s + 6.74·19-s − 11.7·21-s − 23-s + 25-s + 8.12·27-s + 5.21·29-s + 5.95·31-s + 2.83·33-s − 3.95·35-s + 9.12·37-s − 8.12·39-s + 0.252·41-s − 8·43-s + 5.74·45-s − 5.49·47-s + 8.66·49-s + 16.9·51-s − 7.12·53-s + 0.957·55-s + ⋯ |
| L(s) = 1 | + 1.70·3-s + 0.447·5-s − 1.49·7-s + 1.91·9-s + 0.288·11-s − 0.761·13-s + 0.763·15-s + 1.39·17-s + 1.54·19-s − 2.55·21-s − 0.208·23-s + 0.200·25-s + 1.56·27-s + 0.967·29-s + 1.07·31-s + 0.493·33-s − 0.668·35-s + 1.50·37-s − 1.30·39-s + 0.0394·41-s − 1.21·43-s + 0.856·45-s − 0.801·47-s + 1.23·49-s + 2.38·51-s − 0.978·53-s + 0.129·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.211564818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.211564818\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 2.95T + 3T^{2} \) |
| 7 | \( 1 + 3.95T + 7T^{2} \) |
| 11 | \( 1 - 0.957T + 11T^{2} \) |
| 13 | \( 1 + 2.74T + 13T^{2} \) |
| 17 | \( 1 - 5.74T + 17T^{2} \) |
| 19 | \( 1 - 6.74T + 19T^{2} \) |
| 29 | \( 1 - 5.21T + 29T^{2} \) |
| 31 | \( 1 - 5.95T + 31T^{2} \) |
| 37 | \( 1 - 9.12T + 37T^{2} \) |
| 41 | \( 1 - 0.252T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 5.49T + 47T^{2} \) |
| 53 | \( 1 + 7.12T + 53T^{2} \) |
| 59 | \( 1 - 4.78T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 + 9.12T + 67T^{2} \) |
| 71 | \( 1 + 1.66T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 + 0.704T + 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−9.492457685040951349886621166947, −8.479966323743823474100097211405, −7.78835024458228606068362767267, −7.03127122098062682646717401097, −6.25119093457424998147179163532, −5.12926823210431549953248099681, −3.90981480904485413376078697561, −3.03751760170496031134323152463, −2.72278621476486002711094068346, −1.23239733781161442692016829059,
1.23239733781161442692016829059, 2.72278621476486002711094068346, 3.03751760170496031134323152463, 3.90981480904485413376078697561, 5.12926823210431549953248099681, 6.25119093457424998147179163532, 7.03127122098062682646717401097, 7.78835024458228606068362767267, 8.479966323743823474100097211405, 9.492457685040951349886621166947
