| L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s − 13-s + 15-s − 6·17-s − 4·19-s + 2·21-s + 23-s + 25-s − 27-s − 8·29-s − 5·31-s + 2·35-s − 8·37-s + 39-s + 6·41-s + 6·43-s − 45-s + 2·47-s − 3·49-s + 6·51-s + 9·53-s + 4·57-s + 8·59-s + 6·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.277·13-s + 0.258·15-s − 1.45·17-s − 0.917·19-s + 0.436·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s − 1.48·29-s − 0.898·31-s + 0.338·35-s − 1.31·37-s + 0.160·39-s + 0.937·41-s + 0.914·43-s − 0.149·45-s + 0.291·47-s − 3/7·49-s + 0.840·51-s + 1.23·53-s + 0.529·57-s + 1.04·59-s + 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.29994184880875, −12.73998159942772, −12.57425342969415, −11.91340572167888, −11.46064996296146, −10.91998074271270, −10.70588929121708, −10.17033291884164, −9.476904507310723, −9.099381981896481, −8.740684947304652, −8.090624686328295, −7.386947364738086, −7.085186555773336, −6.633775467770994, −6.098333143306047, −5.577621373347175, −5.054930469734112, −4.439792515444477, −3.804213758523338, −3.689532711133328, −2.607682976170428, −2.255271114953238, −1.507205148720996, −0.5096779790858370, 0,
0.5096779790858370, 1.507205148720996, 2.255271114953238, 2.607682976170428, 3.689532711133328, 3.804213758523338, 4.439792515444477, 5.054930469734112, 5.577621373347175, 6.098333143306047, 6.633775467770994, 7.085186555773336, 7.386947364738086, 8.090624686328295, 8.740684947304652, 9.099381981896481, 9.476904507310723, 10.17033291884164, 10.70588929121708, 10.91998074271270, 11.46064996296146, 11.91340572167888, 12.57425342969415, 12.73998159942772, 13.29994184880875
