L(s) = 1 | + 3-s + 4·5-s + 7-s + 9-s − 2·11-s − 6·13-s + 4·15-s − 4·17-s + 4·19-s + 21-s − 2·23-s + 11·25-s + 27-s − 2·29-s − 2·33-s + 4·35-s + 2·37-s − 6·39-s + 4·43-s + 4·45-s − 12·47-s + 49-s − 4·51-s − 6·53-s − 8·55-s + 4·57-s + 8·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.78·5-s + 0.377·7-s + 1/3·9-s − 0.603·11-s − 1.66·13-s + 1.03·15-s − 0.970·17-s + 0.917·19-s + 0.218·21-s − 0.417·23-s + 11/5·25-s + 0.192·27-s − 0.371·29-s − 0.348·33-s + 0.676·35-s + 0.328·37-s − 0.960·39-s + 0.609·43-s + 0.596·45-s − 1.75·47-s + 1/7·49-s − 0.560·51-s − 0.824·53-s − 1.07·55-s + 0.529·57-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.965796601\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.965796601\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−11.50826867908945014059876723308, −10.25657868232859256086223385414, −9.752968916415669329863680305079, −8.977835409861481770724787215218, −7.76556317647643738947138137602, −6.76281206095777722703467003614, −5.54203834680246413327649010589, −4.73251701657473593050786657983, −2.75718031114594033742569839966, −1.92538980972810959214451574073,
1.92538980972810959214451574073, 2.75718031114594033742569839966, 4.73251701657473593050786657983, 5.54203834680246413327649010589, 6.76281206095777722703467003614, 7.76556317647643738947138137602, 8.977835409861481770724787215218, 9.752968916415669329863680305079, 10.25657868232859256086223385414, 11.50826867908945014059876723308