L(s) = 1 | + 3-s + 7-s + 9-s + 6·11-s − 5·13-s − 6·17-s + 5·19-s + 21-s + 6·23-s + 27-s + 6·29-s + 31-s + 6·33-s − 2·37-s − 5·39-s − 43-s − 6·47-s − 6·49-s − 6·51-s + 12·53-s + 5·57-s − 6·59-s + 13·61-s + 63-s + 11·67-s + 6·69-s + 2·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s + 1.80·11-s − 1.38·13-s − 1.45·17-s + 1.14·19-s + 0.218·21-s + 1.25·23-s + 0.192·27-s + 1.11·29-s + 0.179·31-s + 1.04·33-s − 0.328·37-s − 0.800·39-s − 0.152·43-s − 0.875·47-s − 6/7·49-s − 0.840·51-s + 1.64·53-s + 0.662·57-s − 0.781·59-s + 1.66·61-s + 0.125·63-s + 1.34·67-s + 0.722·69-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.815776482\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.815776482\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−8.561613194680960401365058894063, −7.38580951662556061626475687417, −6.97251592485346598623736754257, −6.34671218087642508071319081770, −5.07979857479495636288703767744, −4.61333629110058855638846477864, −3.74053004282009465908034182416, −2.85121436671592400876592597239, −1.96284616627496219343558567397, −0.941088538538383902381260604609,
0.941088538538383902381260604609, 1.96284616627496219343558567397, 2.85121436671592400876592597239, 3.74053004282009465908034182416, 4.61333629110058855638846477864, 5.07979857479495636288703767744, 6.34671218087642508071319081770, 6.97251592485346598623736754257, 7.38580951662556061626475687417, 8.561613194680960401365058894063