| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s + 3·13-s − 14-s + 16-s + 7·17-s + 6·19-s − 9·23-s − 3·26-s + 28-s − 3·29-s − 7·31-s − 32-s − 7·34-s − 10·37-s − 6·38-s + 41-s − 13·43-s + 9·46-s + 2·47-s + 49-s + 3·52-s + 53-s − 56-s + 3·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 0.832·13-s − 0.267·14-s + 1/4·16-s + 1.69·17-s + 1.37·19-s − 1.87·23-s − 0.588·26-s + 0.188·28-s − 0.557·29-s − 1.25·31-s − 0.176·32-s − 1.20·34-s − 1.64·37-s − 0.973·38-s + 0.156·41-s − 1.98·43-s + 1.32·46-s + 0.291·47-s + 1/7·49-s + 0.416·52-s + 0.137·53-s − 0.133·56-s + 0.393·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 381150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 381150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9604628926\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9604628926\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
| 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
−12.17072095847570, −11.96701550677733, −11.71646275197984, −10.94374620818714, −10.70834387936864, −10.12066716625545, −9.798887785379927, −9.360297690711311, −8.824268830619134, −8.365753381205180, −7.805342955721495, −7.661862853018218, −7.133758710912497, −6.499250395318228, −5.968352729150267, −5.459353592834855, −5.291699275916903, −4.448679493547873, −3.687214223996781, −3.452694572275863, −2.972672161933013, −2.044002634207324, −1.483602872900702, −1.328149286408158, −0.2806630867272574,
0.2806630867272574, 1.328149286408158, 1.483602872900702, 2.044002634207324, 2.972672161933013, 3.452694572275863, 3.687214223996781, 4.448679493547873, 5.291699275916903, 5.459353592834855, 5.968352729150267, 6.499250395318228, 7.133758710912497, 7.661862853018218, 7.805342955721495, 8.365753381205180, 8.824268830619134, 9.360297690711311, 9.798887785379927, 10.12066716625545, 10.70834387936864, 10.94374620818714, 11.71646275197984, 11.96701550677733, 12.17072095847570
