| L(s) = 1 | + 1.21·2-s − 0.534·4-s + 5-s − 3.06·8-s + 1.21·10-s + 0.255·11-s − 0.744·13-s − 2.64·16-s − 1.21·17-s + 1.11·19-s − 0.534·20-s + 0.308·22-s + 7.85·23-s + 25-s − 0.901·26-s + 3.32·29-s + 6.91·31-s + 2.93·32-s − 1.46·34-s + 9.27·37-s + 1.34·38-s − 3.06·40-s + 8.81·41-s + 5.70·43-s − 0.136·44-s + 9.51·46-s − 7.91·47-s + ⋯ |
| L(s) = 1 | + 0.856·2-s − 0.267·4-s + 0.447·5-s − 1.08·8-s + 0.382·10-s + 0.0769·11-s − 0.206·13-s − 0.661·16-s − 0.293·17-s + 0.255·19-s − 0.119·20-s + 0.0658·22-s + 1.63·23-s + 0.200·25-s − 0.176·26-s + 0.617·29-s + 1.24·31-s + 0.518·32-s − 0.251·34-s + 1.52·37-s + 0.218·38-s − 0.485·40-s + 1.37·41-s + 0.869·43-s − 0.0205·44-s + 1.40·46-s − 1.15·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.520837804\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.520837804\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 1.21T + 2T^{2} \) |
| 11 | \( 1 - 0.255T + 11T^{2} \) |
| 13 | \( 1 + 0.744T + 13T^{2} \) |
| 17 | \( 1 + 1.21T + 17T^{2} \) |
| 19 | \( 1 - 1.11T + 19T^{2} \) |
| 23 | \( 1 - 7.85T + 23T^{2} \) |
| 29 | \( 1 - 3.32T + 29T^{2} \) |
| 31 | \( 1 - 6.91T + 31T^{2} \) |
| 37 | \( 1 - 9.27T + 37T^{2} \) |
| 41 | \( 1 - 8.81T + 41T^{2} \) |
| 43 | \( 1 - 5.70T + 43T^{2} \) |
| 47 | \( 1 + 7.91T + 47T^{2} \) |
| 53 | \( 1 - 2.42T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 + 9.16T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 + 0.435T + 73T^{2} \) |
| 79 | \( 1 - 2.37T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 - 8.44T + 97T^{2} \) |
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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
−9.231535755540274836483775397587, −8.341819486300324428882040988928, −7.40906548465175031461907283573, −6.38974605804071807883633280116, −5.89738495246158600224745210588, −4.81904205921867415808353726226, −4.48912971122002000963226626181, −3.22884027363687546672121379004, −2.56244390225126283679240940293, −0.941013629185448882805227357705,
0.941013629185448882805227357705, 2.56244390225126283679240940293, 3.22884027363687546672121379004, 4.48912971122002000963226626181, 4.81904205921867415808353726226, 5.89738495246158600224745210588, 6.38974605804071807883633280116, 7.40906548465175031461907283573, 8.341819486300324428882040988928, 9.231535755540274836483775397587
