L(s) = 1 | − 5-s + 7-s + 5·11-s − 2·17-s + 19-s − 23-s − 4·25-s − 4·29-s + 9·31-s − 35-s + 5·37-s + 9·41-s + 10·43-s + 6·47-s + 49-s − 12·53-s − 5·55-s − 14·59-s + 8·67-s − 13·71-s − 2·73-s + 5·77-s − 6·79-s − 4·83-s + 2·85-s + 9·89-s − 95-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.377·7-s + 1.50·11-s − 0.485·17-s + 0.229·19-s − 0.208·23-s − 4/5·25-s − 0.742·29-s + 1.61·31-s − 0.169·35-s + 0.821·37-s + 1.40·41-s + 1.52·43-s + 0.875·47-s + 1/7·49-s − 1.64·53-s − 0.674·55-s − 1.82·59-s + 0.977·67-s − 1.54·71-s − 0.234·73-s + 0.569·77-s − 0.675·79-s − 0.439·83-s + 0.216·85-s + 0.953·89-s − 0.102·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.915352400\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.915352400\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−8.852235364105462314758322151711, −7.82180523610847379974298804156, −7.41731738918212721913056577334, −6.31189353147311880492962393494, −5.91137064624602427944931283698, −4.50854174640145518647490777803, −4.21595600702686973203131728090, −3.16207871637142803917205041928, −1.99414709782610576510843414106, −0.875062595129075836979928350629,
0.875062595129075836979928350629, 1.99414709782610576510843414106, 3.16207871637142803917205041928, 4.21595600702686973203131728090, 4.50854174640145518647490777803, 5.91137064624602427944931283698, 6.31189353147311880492962393494, 7.41731738918212721913056577334, 7.82180523610847379974298804156, 8.852235364105462314758322151711