L(s) = 1 | − 2·2-s + 4·4-s − 5·5-s − 34·7-s − 8·8-s + 10·10-s + 18·11-s − 48·13-s + 68·14-s + 16·16-s + 34·17-s − 19·19-s − 20·20-s − 36·22-s + 128·23-s + 25·25-s + 96·26-s − 136·28-s + 80·29-s + 112·31-s − 32·32-s − 68·34-s + 170·35-s − 124·37-s + 38·38-s + 40·40-s + 208·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.83·7-s − 0.353·8-s + 0.316·10-s + 0.493·11-s − 1.02·13-s + 1.29·14-s + 1/4·16-s + 0.485·17-s − 0.229·19-s − 0.223·20-s − 0.348·22-s + 1.16·23-s + 1/5·25-s + 0.724·26-s − 0.917·28-s + 0.512·29-s + 0.648·31-s − 0.176·32-s − 0.342·34-s + 0.821·35-s − 0.550·37-s + 0.162·38-s + 0.158·40-s + 0.792·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| 19 | \( 1 + p T \) |
good | 7 | \( 1 + 34 T + p^{3} T^{2} \) |
| 11 | \( 1 - 18 T + p^{3} T^{2} \) |
| 13 | \( 1 + 48 T + p^{3} T^{2} \) |
| 17 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 23 | \( 1 - 128 T + p^{3} T^{2} \) |
| 29 | \( 1 - 80 T + p^{3} T^{2} \) |
| 31 | \( 1 - 112 T + p^{3} T^{2} \) |
| 37 | \( 1 + 124 T + p^{3} T^{2} \) |
| 41 | \( 1 - 208 T + p^{3} T^{2} \) |
| 43 | \( 1 - 42 T + p^{3} T^{2} \) |
| 47 | \( 1 - 144 T + p^{3} T^{2} \) |
| 53 | \( 1 - 378 T + p^{3} T^{2} \) |
| 59 | \( 1 + 440 T + p^{3} T^{2} \) |
| 61 | \( 1 + 118 T + p^{3} T^{2} \) |
| 67 | \( 1 - 496 T + p^{3} T^{2} \) |
| 71 | \( 1 + 72 T + p^{3} T^{2} \) |
| 73 | \( 1 + 738 T + p^{3} T^{2} \) |
| 79 | \( 1 - 920 T + p^{3} T^{2} \) |
| 83 | \( 1 + 832 T + p^{3} T^{2} \) |
| 89 | \( 1 + 440 T + p^{3} T^{2} \) |
| 97 | \( 1 + 864 T + p^{3} T^{2} \) |
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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
−8.806937019290670060786224490444, −7.71416407236790916553575183830, −6.97638103806895677898530821546, −6.48379485463422449839038242143, −5.48894379986368456410839592301, −4.24151899065023308387856292028, −3.22549338713940711017747404789, −2.57942163080815858594461788307, −0.951203650861267851357767051002, 0,
0.951203650861267851357767051002, 2.57942163080815858594461788307, 3.22549338713940711017747404789, 4.24151899065023308387856292028, 5.48894379986368456410839592301, 6.48379485463422449839038242143, 6.97638103806895677898530821546, 7.71416407236790916553575183830, 8.806937019290670060786224490444