L(s) = 1 | − 2·7-s + 2·13-s + 17-s + 4·19-s − 6·23-s − 5·25-s + 10·31-s + 8·37-s − 6·41-s + 4·43-s + 12·47-s − 3·49-s − 6·53-s − 12·59-s + 8·61-s + 4·67-s + 6·71-s + 2·73-s + 10·79-s + 12·83-s + 18·89-s − 4·91-s + 14·97-s − 6·101-s + 4·103-s + 20·109-s + 6·113-s + ⋯ |
L(s) = 1 | − 0.755·7-s + 0.554·13-s + 0.242·17-s + 0.917·19-s − 1.25·23-s − 25-s + 1.79·31-s + 1.31·37-s − 0.937·41-s + 0.609·43-s + 1.75·47-s − 3/7·49-s − 0.824·53-s − 1.56·59-s + 1.02·61-s + 0.488·67-s + 0.712·71-s + 0.234·73-s + 1.12·79-s + 1.31·83-s + 1.90·89-s − 0.419·91-s + 1.42·97-s − 0.597·101-s + 0.394·103-s + 1.91·109-s + 0.564·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.608535368\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.608535368\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−9.042520837134499908361226281676, −8.026318567795814427488522879334, −7.59356332550843172899584892694, −6.34124324657337262986339996483, −6.10847916885934502071099583509, −5.00646506423734918535245054397, −3.99639409047484239400750894793, −3.25952156741771651896397782502, −2.20738333359601498441580015494, −0.810408985240977708446989127736,
0.810408985240977708446989127736, 2.20738333359601498441580015494, 3.25952156741771651896397782502, 3.99639409047484239400750894793, 5.00646506423734918535245054397, 6.10847916885934502071099583509, 6.34124324657337262986339996483, 7.59356332550843172899584892694, 8.026318567795814427488522879334, 9.042520837134499908361226281676