L(s) = 1 | − 0.603·2-s − 1.63·4-s + 2.98·5-s + 4.14·7-s + 2.19·8-s − 1.80·10-s + 1.78·11-s − 3.75·13-s − 2.50·14-s + 1.94·16-s − 3.76·17-s + 2.44·19-s − 4.88·20-s − 1.08·22-s − 23-s + 3.90·25-s + 2.26·26-s − 6.77·28-s − 29-s − 0.393·31-s − 5.56·32-s + 2.27·34-s + 12.3·35-s − 0.656·37-s − 1.47·38-s + 6.55·40-s + 7.56·41-s + ⋯ |
L(s) = 1 | − 0.427·2-s − 0.817·4-s + 1.33·5-s + 1.56·7-s + 0.776·8-s − 0.570·10-s + 0.539·11-s − 1.04·13-s − 0.668·14-s + 0.486·16-s − 0.912·17-s + 0.560·19-s − 1.09·20-s − 0.230·22-s − 0.208·23-s + 0.781·25-s + 0.444·26-s − 1.27·28-s − 0.185·29-s − 0.0707·31-s − 0.983·32-s + 0.389·34-s + 2.08·35-s − 0.107·37-s − 0.239·38-s + 1.03·40-s + 1.18·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.099402099\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.099402099\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
good | 2 | \( 1 + 0.603T + 2T^{2} \) |
| 5 | \( 1 - 2.98T + 5T^{2} \) |
| 7 | \( 1 - 4.14T + 7T^{2} \) |
| 11 | \( 1 - 1.78T + 11T^{2} \) |
| 13 | \( 1 + 3.75T + 13T^{2} \) |
| 17 | \( 1 + 3.76T + 17T^{2} \) |
| 19 | \( 1 - 2.44T + 19T^{2} \) |
| 31 | \( 1 + 0.393T + 31T^{2} \) |
| 37 | \( 1 + 0.656T + 37T^{2} \) |
| 41 | \( 1 - 7.56T + 41T^{2} \) |
| 43 | \( 1 - 12.1T + 43T^{2} \) |
| 47 | \( 1 + 12.3T + 47T^{2} \) |
| 53 | \( 1 + 7.63T + 53T^{2} \) |
| 59 | \( 1 - 4.61T + 59T^{2} \) |
| 61 | \( 1 - 2.70T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 - 2.39T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 - 9.97T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 + 0.336T + 89T^{2} \) |
| 97 | \( 1 - 8.25T + 97T^{2} \) |
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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
−8.095884528589636644071803685850, −7.61540753867093082964939937176, −6.70480919772554865934551685808, −5.81211626356069623981114351152, −5.04816548303586333958025900504, −4.73451901530020608428294829948, −3.82342078402134787211852858524, −2.36238010600036818268310934311, −1.81349931888512112484689471248, −0.863315600727074405770138933069,
0.863315600727074405770138933069, 1.81349931888512112484689471248, 2.36238010600036818268310934311, 3.82342078402134787211852858524, 4.73451901530020608428294829948, 5.04816548303586333958025900504, 5.81211626356069623981114351152, 6.70480919772554865934551685808, 7.61540753867093082964939937176, 8.095884528589636644071803685850