| L(s) = 1 | − 1.57·2-s + 0.472·4-s + 1.87·7-s + 2.40·8-s − 1.92·11-s + 0.248·13-s − 2.95·14-s − 4.72·16-s − 7.06·17-s + 19-s + 3.02·22-s + 1.20·23-s − 0.390·26-s + 0.887·28-s − 2.58·29-s + 8.69·31-s + 2.61·32-s + 11.1·34-s − 7.86·37-s − 1.57·38-s − 1.52·41-s + 4.15·43-s − 0.906·44-s − 1.89·46-s + 5.96·47-s − 3.46·49-s + 0.117·52-s + ⋯ |
| L(s) = 1 | − 1.11·2-s + 0.236·4-s + 0.710·7-s + 0.849·8-s − 0.579·11-s + 0.0688·13-s − 0.789·14-s − 1.18·16-s − 1.71·17-s + 0.229·19-s + 0.643·22-s + 0.250·23-s − 0.0765·26-s + 0.167·28-s − 0.479·29-s + 1.56·31-s + 0.462·32-s + 1.90·34-s − 1.29·37-s − 0.255·38-s − 0.238·41-s + 0.633·43-s − 0.136·44-s − 0.278·46-s + 0.870·47-s − 0.495·49-s + 0.0162·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8252223327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8252223327\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 1.57T + 2T^{2} \) |
| 7 | \( 1 - 1.87T + 7T^{2} \) |
| 11 | \( 1 + 1.92T + 11T^{2} \) |
| 13 | \( 1 - 0.248T + 13T^{2} \) |
| 17 | \( 1 + 7.06T + 17T^{2} \) |
| 23 | \( 1 - 1.20T + 23T^{2} \) |
| 29 | \( 1 + 2.58T + 29T^{2} \) |
| 31 | \( 1 - 8.69T + 31T^{2} \) |
| 37 | \( 1 + 7.86T + 37T^{2} \) |
| 41 | \( 1 + 1.52T + 41T^{2} \) |
| 43 | \( 1 - 4.15T + 43T^{2} \) |
| 47 | \( 1 - 5.96T + 47T^{2} \) |
| 53 | \( 1 + 7.11T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 - 5.87T + 61T^{2} \) |
| 67 | \( 1 - 9.53T + 67T^{2} \) |
| 71 | \( 1 - 9.53T + 71T^{2} \) |
| 73 | \( 1 + 6.69T + 73T^{2} \) |
| 79 | \( 1 + 0.348T + 79T^{2} \) |
| 83 | \( 1 + 2.41T + 83T^{2} \) |
| 89 | \( 1 - 17.3T + 89T^{2} \) |
| 97 | \( 1 + 7.70T + 97T^{2} \) |
| 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
−8.452284054637236575937537286340, −7.910420467602668464803462436428, −7.11575969295982271524146112203, −6.49608109441480069470297801643, −5.28545144899920331966970641699, −4.71995339006581868939435248137, −3.91315528730809789098804421088, −2.54872677429678292367003263419, −1.77628854337391536799523297708, −0.61318428255690554187510627659,
0.61318428255690554187510627659, 1.77628854337391536799523297708, 2.54872677429678292367003263419, 3.91315528730809789098804421088, 4.71995339006581868939435248137, 5.28545144899920331966970641699, 6.49608109441480069470297801643, 7.11575969295982271524146112203, 7.910420467602668464803462436428, 8.452284054637236575937537286340
