L(s) = 1 | − 2-s + 3-s + 4-s − 2·5-s − 6-s + 7-s − 8-s + 9-s + 2·10-s + 12-s − 13-s − 14-s − 2·15-s + 16-s + 3·17-s − 18-s − 19-s − 2·20-s + 21-s − 23-s − 24-s − 25-s + 26-s + 27-s + 28-s + 3·29-s + 2·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.288·12-s − 0.277·13-s − 0.267·14-s − 0.516·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.229·19-s − 0.447·20-s + 0.218·21-s − 0.208·23-s − 0.204·24-s − 1/5·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + 0.557·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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
−13.81369736601055, −13.45750316196357, −12.75989844097706, −12.14294831985685, −12.00332442680756, −11.33905088480338, −10.96554061134601, −10.27697449257175, −9.941456794895020, −9.417175682104815, −8.845468583866955, −8.204051448619022, −8.100388100240693, −7.591621847802624, −6.971663863407937, −6.632105360367803, −5.764827207580813, −5.285704096596139, −4.476317878156642, −4.095200385498586, −3.358980397831073, −2.946728511864385, −2.166999502720476, −1.579688373310903, −0.8181048386853851, 0,
0.8181048386853851, 1.579688373310903, 2.166999502720476, 2.946728511864385, 3.358980397831073, 4.095200385498586, 4.476317878156642, 5.285704096596139, 5.764827207580813, 6.632105360367803, 6.971663863407937, 7.591621847802624, 8.100388100240693, 8.204051448619022, 8.845468583866955, 9.417175682104815, 9.941456794895020, 10.27697449257175, 10.96554061134601, 11.33905088480338, 12.00332442680756, 12.14294831985685, 12.75989844097706, 13.45750316196357, 13.81369736601055