| L(s) = 1 | − 5-s + 3.95·7-s + 0.957·11-s − 2.74·13-s − 5.74·17-s − 6.74·19-s − 23-s + 25-s − 5.21·29-s − 5.95·31-s − 3.95·35-s + 9.12·37-s − 0.252·41-s + 8·43-s − 5.49·47-s + 8.66·49-s + 7.12·53-s − 0.957·55-s + 4.78·59-s + 12.4·61-s + 2.74·65-s + 9.12·67-s − 1.66·71-s + 12.3·73-s + 3.78·77-s + 11.8·79-s − 0.704·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.49·7-s + 0.288·11-s − 0.761·13-s − 1.39·17-s − 1.54·19-s − 0.208·23-s + 0.200·25-s − 0.967·29-s − 1.07·31-s − 0.668·35-s + 1.50·37-s − 0.0394·41-s + 1.21·43-s − 0.801·47-s + 1.23·49-s + 0.978·53-s − 0.129·55-s + 0.623·59-s + 1.59·61-s + 0.340·65-s + 1.11·67-s − 0.197·71-s + 1.44·73-s + 0.431·77-s + 1.33·79-s − 0.0773·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.754617271\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.754617271\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 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
−7.88469628691378613844031044835, −7.17949091525851784806766355185, −6.55032331893661723181794685974, −5.63939587331360595410296630255, −4.87198410337445255620268498573, −4.30825237768670758338342987431, −3.77338329867495042223101201360, −2.23155966342735809752659588312, −2.08182565979903314592751751187, −0.63375782700526148436771818713,
0.63375782700526148436771818713, 2.08182565979903314592751751187, 2.23155966342735809752659588312, 3.77338329867495042223101201360, 4.30825237768670758338342987431, 4.87198410337445255620268498573, 5.63939587331360595410296630255, 6.55032331893661723181794685974, 7.17949091525851784806766355185, 7.88469628691378613844031044835
