| L(s) = 1 | − 2.21·2-s + 3-s + 2.90·4-s + 3.81·5-s − 2.21·6-s + 4.67·7-s − 2.00·8-s + 9-s − 8.45·10-s + 1.01·11-s + 2.90·12-s + 13-s − 10.3·14-s + 3.81·15-s − 1.36·16-s + 1.06·17-s − 2.21·18-s + 4.69·19-s + 11.0·20-s + 4.67·21-s − 2.24·22-s − 0.133·23-s − 2.00·24-s + 9.55·25-s − 2.21·26-s + 27-s + 13.5·28-s + ⋯ |
| L(s) = 1 | − 1.56·2-s + 0.577·3-s + 1.45·4-s + 1.70·5-s − 0.904·6-s + 1.76·7-s − 0.709·8-s + 0.333·9-s − 2.67·10-s + 0.304·11-s + 0.838·12-s + 0.277·13-s − 2.76·14-s + 0.985·15-s − 0.341·16-s + 0.258·17-s − 0.522·18-s + 1.07·19-s + 2.47·20-s + 1.02·21-s − 0.477·22-s − 0.0278·23-s − 0.409·24-s + 1.91·25-s − 0.434·26-s + 0.192·27-s + 2.56·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.140067754\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.140067754\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + 2.21T + 2T^{2} \) |
| 5 | \( 1 - 3.81T + 5T^{2} \) |
| 7 | \( 1 - 4.67T + 7T^{2} \) |
| 11 | \( 1 - 1.01T + 11T^{2} \) |
| 17 | \( 1 - 1.06T + 17T^{2} \) |
| 19 | \( 1 - 4.69T + 19T^{2} \) |
| 23 | \( 1 + 0.133T + 23T^{2} \) |
| 29 | \( 1 + 4.34T + 29T^{2} \) |
| 31 | \( 1 + 9.18T + 31T^{2} \) |
| 37 | \( 1 + 9.96T + 37T^{2} \) |
| 41 | \( 1 + 7.56T + 41T^{2} \) |
| 43 | \( 1 - 8.27T + 43T^{2} \) |
| 47 | \( 1 + 4.98T + 47T^{2} \) |
| 53 | \( 1 + 5.95T + 53T^{2} \) |
| 59 | \( 1 - 3.35T + 59T^{2} \) |
| 61 | \( 1 - 7.55T + 61T^{2} \) |
| 67 | \( 1 + 3.22T + 67T^{2} \) |
| 71 | \( 1 - 7.00T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 - 6.15T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 - 6.38T + 97T^{2} \) |
| 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
−8.779688366589890391853107138888, −7.83423815967623714994584941821, −7.35588009892404739797075883075, −6.51861069469162108136410673634, −5.43382499455042661412401274511, −5.01245677002497351431064305520, −3.60125514927423061241230846875, −2.24520655697199050008776730677, −1.75329845696491408584074173758, −1.19212022198860680400689993491,
1.19212022198860680400689993491, 1.75329845696491408584074173758, 2.24520655697199050008776730677, 3.60125514927423061241230846875, 5.01245677002497351431064305520, 5.43382499455042661412401274511, 6.51861069469162108136410673634, 7.35588009892404739797075883075, 7.83423815967623714994584941821, 8.779688366589890391853107138888
