| L(s) = 1 | + 2.29·2-s − 0.141·3-s + 3.25·4-s + 2.10·5-s − 0.324·6-s + 3.17·7-s + 2.88·8-s − 2.97·9-s + 4.82·10-s + 3.76·11-s − 0.460·12-s − 2.55·13-s + 7.28·14-s − 0.297·15-s + 0.0955·16-s − 17-s − 6.83·18-s − 3.88·19-s + 6.85·20-s − 0.449·21-s + 8.63·22-s + 5.00·23-s − 0.407·24-s − 0.570·25-s − 5.86·26-s + 0.846·27-s + 10.3·28-s + ⋯ |
| L(s) = 1 | + 1.62·2-s − 0.0817·3-s + 1.62·4-s + 0.941·5-s − 0.132·6-s + 1.20·7-s + 1.01·8-s − 0.993·9-s + 1.52·10-s + 1.13·11-s − 0.133·12-s − 0.709·13-s + 1.94·14-s − 0.0769·15-s + 0.0238·16-s − 0.242·17-s − 1.61·18-s − 0.892·19-s + 1.53·20-s − 0.0981·21-s + 1.84·22-s + 1.04·23-s − 0.0832·24-s − 0.114·25-s − 1.14·26-s + 0.162·27-s + 1.95·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.636001195\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.636001195\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 2 | \( 1 - 2.29T + 2T^{2} \) |
| 3 | \( 1 + 0.141T + 3T^{2} \) |
| 5 | \( 1 - 2.10T + 5T^{2} \) |
| 7 | \( 1 - 3.17T + 7T^{2} \) |
| 11 | \( 1 - 3.76T + 11T^{2} \) |
| 13 | \( 1 + 2.55T + 13T^{2} \) |
| 19 | \( 1 + 3.88T + 19T^{2} \) |
| 23 | \( 1 - 5.00T + 23T^{2} \) |
| 29 | \( 1 + 2.31T + 29T^{2} \) |
| 31 | \( 1 - 9.70T + 31T^{2} \) |
| 37 | \( 1 + 11.7T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 5.52T + 43T^{2} \) |
| 47 | \( 1 + 1.04T + 47T^{2} \) |
| 53 | \( 1 - 1.97T + 53T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 - 6.08T + 67T^{2} \) |
| 71 | \( 1 + 8.89T + 71T^{2} \) |
| 73 | \( 1 - 14.2T + 73T^{2} \) |
| 79 | \( 1 + 3.41T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 - 1.04T + 89T^{2} \) |
| 97 | \( 1 + 0.00580T + 97T^{2} \) |
| 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
−10.21469782538156626707578842178, −9.046728326960312053844475708169, −8.352862265733427113778347748653, −6.93962932195053082283415520613, −6.34621707957431711586142335480, −5.36892661241692879845186523908, −4.91920279666988466967667694677, −3.89229245178744997814947050703, −2.66170862534039660419982566127, −1.76825042755952950431229386245,
1.76825042755952950431229386245, 2.66170862534039660419982566127, 3.89229245178744997814947050703, 4.91920279666988466967667694677, 5.36892661241692879845186523908, 6.34621707957431711586142335480, 6.93962932195053082283415520613, 8.352862265733427113778347748653, 9.046728326960312053844475708169, 10.21469782538156626707578842178
