L(s) = 1 | + 2-s + 4-s − 7-s + 8-s + 4·11-s + 2·13-s − 14-s + 16-s − 6·17-s + 4·19-s + 4·22-s − 23-s + 2·26-s − 28-s + 2·29-s − 8·31-s + 32-s − 6·34-s − 6·37-s + 4·38-s + 6·41-s + 4·43-s + 4·44-s − 46-s − 8·47-s + 49-s + 2·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 1.20·11-s + 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.917·19-s + 0.852·22-s − 0.208·23-s + 0.392·26-s − 0.188·28-s + 0.371·29-s − 1.43·31-s + 0.176·32-s − 1.02·34-s − 0.986·37-s + 0.648·38-s + 0.937·41-s + 0.609·43-s + 0.603·44-s − 0.147·46-s − 1.16·47-s + 1/7·49-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−14.12117741582448, −13.92503543100214, −13.47567347955242, −12.74298396279184, −12.52925294408867, −11.89009406701758, −11.29562098756632, −11.09774998829028, −10.46834145504364, −9.744482160766805, −9.268198765280542, −8.832092038588604, −8.301294094077563, −7.385396832228009, −7.118561835173480, −6.447644992714732, −6.100319358881915, −5.499138613113521, −4.794264442945736, −4.235954944323767, −3.698824204073866, −3.249012962505881, −2.441393216910755, −1.745189053376929, −1.101733600105433, 0,
1.101733600105433, 1.745189053376929, 2.441393216910755, 3.249012962505881, 3.698824204073866, 4.235954944323767, 4.794264442945736, 5.499138613113521, 6.100319358881915, 6.447644992714732, 7.118561835173480, 7.385396832228009, 8.301294094077563, 8.832092038588604, 9.268198765280542, 9.744482160766805, 10.46834145504364, 11.09774998829028, 11.29562098756632, 11.89009406701758, 12.52925294408867, 12.74298396279184, 13.47567347955242, 13.92503543100214, 14.12117741582448