| L(s) = 1 | + 3-s − 5-s − 2.80·7-s + 9-s + 0.770·11-s − 3.59·13-s − 15-s − 0.618·17-s − 6.93·19-s − 2.80·21-s + 1.25·23-s + 25-s + 27-s + 3.30·29-s + 3.43·31-s + 0.770·33-s + 2.80·35-s − 5.13·37-s − 3.59·39-s − 1.43·41-s − 4.84·43-s − 45-s + 10.8·47-s + 0.840·49-s − 0.618·51-s + 12.6·53-s − 0.770·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.05·7-s + 0.333·9-s + 0.232·11-s − 0.995·13-s − 0.258·15-s − 0.149·17-s − 1.59·19-s − 0.611·21-s + 0.261·23-s + 0.200·25-s + 0.192·27-s + 0.613·29-s + 0.616·31-s + 0.134·33-s + 0.473·35-s − 0.843·37-s − 0.574·39-s − 0.223·41-s − 0.738·43-s − 0.149·45-s + 1.58·47-s + 0.120·49-s − 0.0865·51-s + 1.74·53-s − 0.103·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.362415592\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.362415592\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| good | 7 | \( 1 + 2.80T + 7T^{2} \) |
| 11 | \( 1 - 0.770T + 11T^{2} \) |
| 13 | \( 1 + 3.59T + 13T^{2} \) |
| 17 | \( 1 + 0.618T + 17T^{2} \) |
| 19 | \( 1 + 6.93T + 19T^{2} \) |
| 23 | \( 1 - 1.25T + 23T^{2} \) |
| 29 | \( 1 - 3.30T + 29T^{2} \) |
| 31 | \( 1 - 3.43T + 31T^{2} \) |
| 37 | \( 1 + 5.13T + 37T^{2} \) |
| 41 | \( 1 + 1.43T + 41T^{2} \) |
| 43 | \( 1 + 4.84T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 - 0.397T + 59T^{2} \) |
| 61 | \( 1 + 12.9T + 61T^{2} \) |
| 71 | \( 1 + 1.10T + 71T^{2} \) |
| 73 | \( 1 + 2.18T + 73T^{2} \) |
| 79 | \( 1 - 9.90T + 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 - 18.2T + 97T^{2} \) |
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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
−7.81120253275943434040387587568, −7.08106593339965020959308674593, −6.63630888328890681323398488943, −5.89334191035943005024976557210, −4.81658561376792760534465004097, −4.23740032555640320611370875275, −3.44435859095274915156808371700, −2.73900783598996625954076472781, −1.98144757096651863868258708814, −0.53318340533996517484602872159,
0.53318340533996517484602872159, 1.98144757096651863868258708814, 2.73900783598996625954076472781, 3.44435859095274915156808371700, 4.23740032555640320611370875275, 4.81658561376792760534465004097, 5.89334191035943005024976557210, 6.63630888328890681323398488943, 7.08106593339965020959308674593, 7.81120253275943434040387587568
