| L(s) = 1 | − 2.30·5-s − 3.57·7-s − 4.44·11-s − 0.354·13-s + 3.59·17-s − 7.12·19-s + 1.95·23-s + 0.319·25-s + 8.39·29-s − 9.47·31-s + 8.25·35-s + 37-s − 6.69·41-s + 11.1·43-s + 0.457·47-s + 5.80·49-s − 5.80·53-s + 10.2·55-s + 15.0·59-s + 2·61-s + 0.818·65-s − 13.0·67-s + 0.457·71-s + 10.1·73-s + 15.8·77-s + 0.765·79-s + 16.6·83-s + ⋯ |
| L(s) = 1 | − 1.03·5-s − 1.35·7-s − 1.33·11-s − 0.0983·13-s + 0.870·17-s − 1.63·19-s + 0.406·23-s + 0.0639·25-s + 1.55·29-s − 1.70·31-s + 1.39·35-s + 0.164·37-s − 1.04·41-s + 1.70·43-s + 0.0667·47-s + 0.828·49-s − 0.796·53-s + 1.38·55-s + 1.95·59-s + 0.256·61-s + 0.101·65-s − 1.59·67-s + 0.0542·71-s + 1.19·73-s + 1.81·77-s + 0.0861·79-s + 1.83·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6126548497\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6126548497\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 5 | \( 1 + 2.30T + 5T^{2} \) |
| 7 | \( 1 + 3.57T + 7T^{2} \) |
| 11 | \( 1 + 4.44T + 11T^{2} \) |
| 13 | \( 1 + 0.354T + 13T^{2} \) |
| 17 | \( 1 - 3.59T + 17T^{2} \) |
| 19 | \( 1 + 7.12T + 19T^{2} \) |
| 23 | \( 1 - 1.95T + 23T^{2} \) |
| 29 | \( 1 - 8.39T + 29T^{2} \) |
| 31 | \( 1 + 9.47T + 31T^{2} \) |
| 41 | \( 1 + 6.69T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 0.457T + 47T^{2} \) |
| 53 | \( 1 + 5.80T + 53T^{2} \) |
| 59 | \( 1 - 15.0T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 13.0T + 67T^{2} \) |
| 71 | \( 1 - 0.457T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 - 0.765T + 79T^{2} \) |
| 83 | \( 1 - 16.6T + 83T^{2} \) |
| 89 | \( 1 + 6.28T + 89T^{2} \) |
| 97 | \( 1 - 6.70T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.753995375998617393296683679389, −8.047525396905369996546494346601, −7.39366525640116361737925393924, −6.63073193405294156873150670043, −5.81733351637056639973057950910, −4.89633653025997335320013620917, −3.92086265269222323941865138427, −3.22610152493788464115476599046, −2.35130605811605934234486712290, −0.45859705460120520279036375400,
0.45859705460120520279036375400, 2.35130605811605934234486712290, 3.22610152493788464115476599046, 3.92086265269222323941865138427, 4.89633653025997335320013620917, 5.81733351637056639973057950910, 6.63073193405294156873150670043, 7.39366525640116361737925393924, 8.047525396905369996546494346601, 8.753995375998617393296683679389
