L(s) = 1 | − 5-s + 7-s − 7·13-s + 4·17-s − 19-s + 25-s − 29-s + 3·31-s − 35-s + 8·37-s − 9·41-s + 43-s − 10·47-s − 6·49-s + 14·53-s − 5·61-s + 7·65-s − 11·67-s + 10·71-s + 7·73-s + 13·79-s − 8·83-s − 4·85-s + 12·89-s − 7·91-s + 95-s + 2·97-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.377·7-s − 1.94·13-s + 0.970·17-s − 0.229·19-s + 1/5·25-s − 0.185·29-s + 0.538·31-s − 0.169·35-s + 1.31·37-s − 1.40·41-s + 0.152·43-s − 1.45·47-s − 6/7·49-s + 1.92·53-s − 0.640·61-s + 0.868·65-s − 1.34·67-s + 1.18·71-s + 0.819·73-s + 1.46·79-s − 0.878·83-s − 0.433·85-s + 1.27·89-s − 0.733·91-s + 0.102·95-s + 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 43 | \( 1 - T \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−13.79525016341386, −13.25934188735257, −12.74685774267755, −12.19091678860267, −11.91833428769566, −11.51224114768525, −10.91165971383142, −10.18190285388446, −10.05045597464141, −9.411335979865307, −8.966375353481826, −8.111076038186750, −7.908783674836635, −7.505866642042541, −6.787032225300702, −6.476454171605979, −5.578900933082432, −5.130592999204526, −4.723789930778515, −4.155026193897734, −3.454709543703420, −2.866022710624240, −2.304862512477215, −1.616347695203834, −0.7677882734084680, 0,
0.7677882734084680, 1.616347695203834, 2.304862512477215, 2.866022710624240, 3.454709543703420, 4.155026193897734, 4.723789930778515, 5.130592999204526, 5.578900933082432, 6.476454171605979, 6.787032225300702, 7.505866642042541, 7.908783674836635, 8.111076038186750, 8.966375353481826, 9.411335979865307, 10.05045597464141, 10.18190285388446, 10.91165971383142, 11.51224114768525, 11.91833428769566, 12.19091678860267, 12.74685774267755, 13.25934188735257, 13.79525016341386