| L(s) = 1 | − 0.193·2-s + 1.67·3-s − 1.96·4-s − 0.806·5-s − 0.324·6-s + 0.324·7-s + 0.768·8-s − 0.193·9-s + 0.156·10-s − 4.63·11-s − 3.28·12-s + 2.96·13-s − 0.0630·14-s − 1.35·15-s + 3.77·16-s − 2·17-s + 0.0376·18-s + 6.63·19-s + 1.58·20-s + 0.544·21-s + 0.899·22-s + 8.31·23-s + 1.28·24-s − 4.35·25-s − 0.574·26-s − 5.35·27-s − 0.637·28-s + ⋯ |
| L(s) = 1 | − 0.137·2-s + 0.967·3-s − 0.981·4-s − 0.360·5-s − 0.132·6-s + 0.122·7-s + 0.271·8-s − 0.0646·9-s + 0.0494·10-s − 1.39·11-s − 0.948·12-s + 0.821·13-s − 0.0168·14-s − 0.348·15-s + 0.943·16-s − 0.485·17-s + 0.00886·18-s + 1.52·19-s + 0.353·20-s + 0.118·21-s + 0.191·22-s + 1.73·23-s + 0.262·24-s − 0.870·25-s − 0.112·26-s − 1.02·27-s − 0.120·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7593268570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7593268570\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 41 | \( 1 - T \) |
| good | 2 | \( 1 + 0.193T + 2T^{2} \) |
| 3 | \( 1 - 1.67T + 3T^{2} \) |
| 5 | \( 1 + 0.806T + 5T^{2} \) |
| 7 | \( 1 - 0.324T + 7T^{2} \) |
| 11 | \( 1 + 4.63T + 11T^{2} \) |
| 13 | \( 1 - 2.96T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 6.63T + 19T^{2} \) |
| 23 | \( 1 - 8.31T + 23T^{2} \) |
| 29 | \( 1 + 5.35T + 29T^{2} \) |
| 31 | \( 1 - 5.61T + 31T^{2} \) |
| 37 | \( 1 + 2.41T + 37T^{2} \) |
| 43 | \( 1 + 4.96T + 43T^{2} \) |
| 47 | \( 1 + 5.86T + 47T^{2} \) |
| 53 | \( 1 + 1.35T + 53T^{2} \) |
| 59 | \( 1 - 4.31T + 59T^{2} \) |
| 61 | \( 1 - 4.57T + 61T^{2} \) |
| 67 | \( 1 - 4.63T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 - 8.71T + 79T^{2} \) |
| 83 | \( 1 + 6.70T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 - 8.70T + 97T^{2} \) |
| 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
−15.93455743675793656818101393405, −14.91336200523809483351601555541, −13.66946044047958969524487815118, −13.13170536541987678948922981983, −11.26569872319727172613840710601, −9.719343105570443308267388550378, −8.583196060793129675853966373770, −7.71870526716192346254788143179, −5.19303412010009912893951517847, −3.33309696303425523695901098260,
3.33309696303425523695901098260, 5.19303412010009912893951517847, 7.71870526716192346254788143179, 8.583196060793129675853966373770, 9.719343105570443308267388550378, 11.26569872319727172613840710601, 13.13170536541987678948922981983, 13.66946044047958969524487815118, 14.91336200523809483351601555541, 15.93455743675793656818101393405
