L(s) = 1 | − 7-s + 4·11-s − 2·13-s + 4·17-s − 4·23-s − 5·25-s + 4·29-s + 8·31-s + 2·37-s − 4·41-s + 8·43-s − 8·47-s + 49-s − 4·53-s + 8·59-s + 2·61-s + 8·67-s − 12·71-s + 6·73-s − 4·77-s + 8·79-s + 16·83-s − 12·89-s + 2·91-s − 2·97-s + 8·103-s + 4·107-s + ⋯ |
L(s) = 1 | − 0.377·7-s + 1.20·11-s − 0.554·13-s + 0.970·17-s − 0.834·23-s − 25-s + 0.742·29-s + 1.43·31-s + 0.328·37-s − 0.624·41-s + 1.21·43-s − 1.16·47-s + 1/7·49-s − 0.549·53-s + 1.04·59-s + 0.256·61-s + 0.977·67-s − 1.42·71-s + 0.702·73-s − 0.455·77-s + 0.900·79-s + 1.75·83-s − 1.27·89-s + 0.209·91-s − 0.203·97-s + 0.788·103-s + 0.386·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.888429370\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.888429370\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.312800947259352999029654855028, −7.82657075768956775624561038408, −6.85455547345968172173263225562, −6.30976897034687746205351939386, −5.56893022235200886998375054341, −4.57255409210211494302382224210, −3.85337102430247200397131844284, −3.01871027790884328597915787157, −1.95048060242068047806603121302, −0.799812534367287661440970892005,
0.799812534367287661440970892005, 1.95048060242068047806603121302, 3.01871027790884328597915787157, 3.85337102430247200397131844284, 4.57255409210211494302382224210, 5.56893022235200886998375054341, 6.30976897034687746205351939386, 6.85455547345968172173263225562, 7.82657075768956775624561038408, 8.312800947259352999029654855028