L(s) = 1 | + 5-s − 7-s + 2·13-s − 6·17-s + 4·19-s − 23-s + 25-s − 6·29-s + 4·31-s − 35-s + 2·37-s + 6·41-s − 8·43-s + 12·47-s + 49-s + 6·53-s − 12·59-s + 2·61-s + 2·65-s − 8·67-s + 2·73-s − 8·79-s + 12·83-s − 6·85-s − 6·89-s − 2·91-s + 4·95-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.377·7-s + 0.554·13-s − 1.45·17-s + 0.917·19-s − 0.208·23-s + 1/5·25-s − 1.11·29-s + 0.718·31-s − 0.169·35-s + 0.328·37-s + 0.937·41-s − 1.21·43-s + 1.75·47-s + 1/7·49-s + 0.824·53-s − 1.56·59-s + 0.256·61-s + 0.248·65-s − 0.977·67-s + 0.234·73-s − 0.900·79-s + 1.31·83-s − 0.650·85-s − 0.635·89-s − 0.209·91-s + 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.105399131\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.105399131\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.56869280830700, −13.30182937198095, −12.68243581963641, −12.15617664289424, −11.68577907563822, −11.01770527429545, −10.84521646139164, −10.13392654895327, −9.659079652556586, −9.186559991663752, −8.784478284405266, −8.275234361439006, −7.461197798246447, −7.240214057229371, −6.432629822340072, −6.148762833096527, −5.575883681603004, −5.010205729025273, −4.291337916865864, −3.909751635341366, −3.124062581575114, −2.603911080808049, −1.958282904610860, −1.269951854769936, −0.4545681757956093,
0.4545681757956093, 1.269951854769936, 1.958282904610860, 2.603911080808049, 3.124062581575114, 3.909751635341366, 4.291337916865864, 5.010205729025273, 5.575883681603004, 6.148762833096527, 6.432629822340072, 7.240214057229371, 7.461197798246447, 8.275234361439006, 8.784478284405266, 9.186559991663752, 9.659079652556586, 10.13392654895327, 10.84521646139164, 11.01770527429545, 11.68577907563822, 12.15617664289424, 12.68243581963641, 13.30182937198095, 13.56869280830700