L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s + 11-s + 4·13-s − 14-s + 16-s + 4·17-s + 8·19-s − 20-s − 22-s − 2·23-s + 25-s − 4·26-s + 28-s + 6·29-s − 4·31-s − 32-s − 4·34-s − 35-s + 4·37-s − 8·38-s + 40-s − 2·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s − 0.353·8-s + 0.316·10-s + 0.301·11-s + 1.10·13-s − 0.267·14-s + 1/4·16-s + 0.970·17-s + 1.83·19-s − 0.223·20-s − 0.213·22-s − 0.417·23-s + 1/5·25-s − 0.784·26-s + 0.188·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.685·34-s − 0.169·35-s + 0.657·37-s − 1.29·38-s + 0.158·40-s − 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.752626305\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.752626305\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.106528583424502003860753476548, −7.36687005204005099315916852421, −6.78557820012707753594936407152, −5.81845082523083154762403164471, −5.34127513415065276753022266131, −4.21423836318721245574563258973, −3.49962045972582161365173405578, −2.74168068907712170365109578034, −1.45019246556085374202748376321, −0.850669001930898295528103175747,
0.850669001930898295528103175747, 1.45019246556085374202748376321, 2.74168068907712170365109578034, 3.49962045972582161365173405578, 4.21423836318721245574563258973, 5.34127513415065276753022266131, 5.81845082523083154762403164471, 6.78557820012707753594936407152, 7.36687005204005099315916852421, 8.106528583424502003860753476548