L(s) = 1 | + 2.41·2-s + 3.82·4-s + 5-s − 2.82·7-s + 4.41·8-s + 2.41·10-s + 0.414·11-s − 3.82·13-s − 6.82·14-s + 2.99·16-s − 0.828·17-s + 6·19-s + 3.82·20-s + 0.999·22-s − 3.65·23-s − 4·25-s − 9.24·26-s − 10.8·28-s − 29-s + 10.0·31-s − 1.58·32-s − 1.99·34-s − 2.82·35-s − 4·37-s + 14.4·38-s + 4.41·40-s + 4.48·41-s + ⋯ |
L(s) = 1 | + 1.70·2-s + 1.91·4-s + 0.447·5-s − 1.06·7-s + 1.56·8-s + 0.763·10-s + 0.124·11-s − 1.06·13-s − 1.82·14-s + 0.749·16-s − 0.200·17-s + 1.37·19-s + 0.856·20-s + 0.213·22-s − 0.762·23-s − 0.800·25-s − 1.81·26-s − 2.04·28-s − 0.185·29-s + 1.80·31-s − 0.280·32-s − 0.342·34-s − 0.478·35-s − 0.657·37-s + 2.34·38-s + 0.697·40-s + 0.700·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.925953774\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.925953774\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 29 | \( 1 + T \) |
good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 - 0.414T + 11T^{2} \) |
| 13 | \( 1 + 3.82T + 13T^{2} \) |
| 17 | \( 1 + 0.828T + 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + 3.65T + 23T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 - 4.48T + 41T^{2} \) |
| 43 | \( 1 - 3.58T + 43T^{2} \) |
| 47 | \( 1 - 3.24T + 47T^{2} \) |
| 53 | \( 1 + 9.48T + 53T^{2} \) |
| 59 | \( 1 - 3.65T + 59T^{2} \) |
| 61 | \( 1 + 4.82T + 61T^{2} \) |
| 67 | \( 1 - 5.65T + 67T^{2} \) |
| 71 | \( 1 - 8.82T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 2.41T + 79T^{2} \) |
| 83 | \( 1 + 7.65T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 - 4.48T + 97T^{2} \) |
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
−12.23621555334175947924322218029, −11.53243840575287084719496578146, −10.13849306395019046040820746828, −9.430617703393691399289608024241, −7.61443142486533455472104152675, −6.56721286853034092137906157602, −5.79406226310636244204274697138, −4.74002701089451450913131013016, −3.50102159378500060775047755493, −2.44159806729669538613903024767,
2.44159806729669538613903024767, 3.50102159378500060775047755493, 4.74002701089451450913131013016, 5.79406226310636244204274697138, 6.56721286853034092137906157602, 7.61443142486533455472104152675, 9.430617703393691399289608024241, 10.13849306395019046040820746828, 11.53243840575287084719496578146, 12.23621555334175947924322218029