L(s) = 1 | + 2.16·3-s − 0.502·5-s + 7-s + 1.68·9-s − 3.41·11-s + 4.73·13-s − 1.08·15-s + 6.89·17-s + 7.51·19-s + 2.16·21-s + 23-s − 4.74·25-s − 2.84·27-s − 0.637·29-s + 5.23·31-s − 7.39·33-s − 0.502·35-s − 5.84·37-s + 10.2·39-s − 5.06·41-s + 4.37·43-s − 0.847·45-s − 11.6·47-s + 49-s + 14.9·51-s + 2.12·53-s + 1.71·55-s + ⋯ |
L(s) = 1 | + 1.24·3-s − 0.224·5-s + 0.377·7-s + 0.562·9-s − 1.03·11-s + 1.31·13-s − 0.280·15-s + 1.67·17-s + 1.72·19-s + 0.472·21-s + 0.208·23-s − 0.949·25-s − 0.547·27-s − 0.118·29-s + 0.940·31-s − 1.28·33-s − 0.0849·35-s − 0.960·37-s + 1.64·39-s − 0.790·41-s + 0.667·43-s − 0.126·45-s − 1.69·47-s + 0.142·49-s + 2.09·51-s + 0.292·53-s + 0.231·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.284963151\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.284963151\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 - 2.16T + 3T^{2} \) |
| 5 | \( 1 + 0.502T + 5T^{2} \) |
| 11 | \( 1 + 3.41T + 11T^{2} \) |
| 13 | \( 1 - 4.73T + 13T^{2} \) |
| 17 | \( 1 - 6.89T + 17T^{2} \) |
| 19 | \( 1 - 7.51T + 19T^{2} \) |
| 29 | \( 1 + 0.637T + 29T^{2} \) |
| 31 | \( 1 - 5.23T + 31T^{2} \) |
| 37 | \( 1 + 5.84T + 37T^{2} \) |
| 41 | \( 1 + 5.06T + 41T^{2} \) |
| 43 | \( 1 - 4.37T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 - 2.12T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 + 1.36T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 - 14.0T + 73T^{2} \) |
| 79 | \( 1 + 16.2T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 - 4.59T + 89T^{2} \) |
| 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
−10.37981477821821275672867968948, −9.628360181180263669642654695907, −8.691665478611334922003142083666, −7.85839635152216424126381495334, −7.59432590223243088640573990213, −5.97210294760431702358640811434, −5.03867829476756277004510669128, −3.55041914879140294850863792302, −3.03557169392546915624058374793, −1.48176093926026568115985525678,
1.48176093926026568115985525678, 3.03557169392546915624058374793, 3.55041914879140294850863792302, 5.03867829476756277004510669128, 5.97210294760431702358640811434, 7.59432590223243088640573990213, 7.85839635152216424126381495334, 8.691665478611334922003142083666, 9.628360181180263669642654695907, 10.37981477821821275672867968948