L(s) = 1 | + 2.72·2-s − 2.49·3-s + 5.44·4-s + 3.31·5-s − 6.81·6-s − 3.55·7-s + 9.41·8-s + 3.23·9-s + 9.05·10-s + 0.514·11-s − 13.6·12-s + 4.65·13-s − 9.69·14-s − 8.29·15-s + 14.7·16-s + 5.82·17-s + 8.83·18-s − 2.48·19-s + 18.0·20-s + 8.87·21-s + 1.40·22-s − 1.78·23-s − 23.5·24-s + 6.02·25-s + 12.7·26-s − 0.596·27-s − 19.3·28-s + ⋯ |
L(s) = 1 | + 1.92·2-s − 1.44·3-s + 2.72·4-s + 1.48·5-s − 2.78·6-s − 1.34·7-s + 3.32·8-s + 1.07·9-s + 2.86·10-s + 0.155·11-s − 3.92·12-s + 1.29·13-s − 2.59·14-s − 2.14·15-s + 3.69·16-s + 1.41·17-s + 2.08·18-s − 0.569·19-s + 4.04·20-s + 1.93·21-s + 0.299·22-s − 0.371·23-s − 4.79·24-s + 1.20·25-s + 2.49·26-s − 0.114·27-s − 3.65·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.065220760\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.065220760\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2671 | \( 1 - T \) |
good | 2 | \( 1 - 2.72T + 2T^{2} \) |
| 3 | \( 1 + 2.49T + 3T^{2} \) |
| 5 | \( 1 - 3.31T + 5T^{2} \) |
| 7 | \( 1 + 3.55T + 7T^{2} \) |
| 11 | \( 1 - 0.514T + 11T^{2} \) |
| 13 | \( 1 - 4.65T + 13T^{2} \) |
| 17 | \( 1 - 5.82T + 17T^{2} \) |
| 19 | \( 1 + 2.48T + 19T^{2} \) |
| 23 | \( 1 + 1.78T + 23T^{2} \) |
| 29 | \( 1 + 6.80T + 29T^{2} \) |
| 31 | \( 1 + 9.48T + 31T^{2} \) |
| 37 | \( 1 - 1.45T + 37T^{2} \) |
| 41 | \( 1 - 2.28T + 41T^{2} \) |
| 43 | \( 1 - 9.33T + 43T^{2} \) |
| 47 | \( 1 - 4.17T + 47T^{2} \) |
| 53 | \( 1 - 9.66T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 + 7.34T + 61T^{2} \) |
| 67 | \( 1 - 3.28T + 67T^{2} \) |
| 71 | \( 1 + 2.56T + 71T^{2} \) |
| 73 | \( 1 + 6.11T + 73T^{2} \) |
| 79 | \( 1 - 8.87T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 - 1.29T + 89T^{2} \) |
| 97 | \( 1 + 1.55T + 97T^{2} \) |
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
−9.116355031857156823301259049378, −7.43618489508150352460118354866, −6.67842769517727428591821042457, −6.08324226129943624803490617730, −5.64403585019133213334386796124, −5.48953995824004704381195062748, −4.10307957369195350821103440621, −3.47242472751702715661800271562, −2.34920101777494267688644869822, −1.25403905410333616252879060476,
1.25403905410333616252879060476, 2.34920101777494267688644869822, 3.47242472751702715661800271562, 4.10307957369195350821103440621, 5.48953995824004704381195062748, 5.64403585019133213334386796124, 6.08324226129943624803490617730, 6.67842769517727428591821042457, 7.43618489508150352460118354866, 9.116355031857156823301259049378