L(s) = 1 | − 7-s − 4·11-s − 6·13-s + 2·17-s + 6·29-s − 8·31-s − 10·37-s − 2·41-s − 4·43-s − 8·47-s + 49-s + 2·53-s + 8·59-s + 14·61-s + 12·67-s − 16·71-s − 2·73-s + 4·77-s + 8·79-s + 8·83-s − 10·89-s + 6·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
L(s) = 1 | − 0.377·7-s − 1.20·11-s − 1.66·13-s + 0.485·17-s + 1.11·29-s − 1.43·31-s − 1.64·37-s − 0.312·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s + 0.274·53-s + 1.04·59-s + 1.79·61-s + 1.46·67-s − 1.89·71-s − 0.234·73-s + 0.455·77-s + 0.900·79-s + 0.878·83-s − 1.05·89-s + 0.628·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.05413105707264, −13.38819287227690, −12.99190291196859, −12.52860115445543, −12.14346819688546, −11.61541591257306, −11.05211757581408, −10.28325269831507, −10.13796078762018, −9.756410442213990, −9.014303799450845, −8.503798932292613, −7.949524993358179, −7.487146094781965, −6.878233161411640, −6.661723425327035, −5.566150242221185, −5.343341702820986, −4.903576252916919, −4.200689543641700, −3.373765011156887, −3.018779922786098, −2.256722815081749, −1.837507404427671, −0.6599961150823585, 0,
0.6599961150823585, 1.837507404427671, 2.256722815081749, 3.018779922786098, 3.373765011156887, 4.200689543641700, 4.903576252916919, 5.343341702820986, 5.566150242221185, 6.661723425327035, 6.878233161411640, 7.487146094781965, 7.949524993358179, 8.503798932292613, 9.014303799450845, 9.756410442213990, 10.13796078762018, 10.28325269831507, 11.05211757581408, 11.61541591257306, 12.14346819688546, 12.52860115445543, 12.99190291196859, 13.38819287227690, 14.05413105707264