| L(s) = 1 | − 3·5-s + 3·11-s + 2·13-s + 6·17-s − 2·19-s − 6·23-s + 4·25-s + 9·29-s − 7·31-s + 10·37-s − 4·43-s − 12·47-s − 3·53-s − 9·55-s − 3·59-s − 4·61-s − 6·65-s + 2·67-s − 2·73-s − 5·79-s + 9·83-s − 18·85-s − 6·89-s + 6·95-s + 13·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.904·11-s + 0.554·13-s + 1.45·17-s − 0.458·19-s − 1.25·23-s + 4/5·25-s + 1.67·29-s − 1.25·31-s + 1.64·37-s − 0.609·43-s − 1.75·47-s − 0.412·53-s − 1.21·55-s − 0.390·59-s − 0.512·61-s − 0.744·65-s + 0.244·67-s − 0.234·73-s − 0.562·79-s + 0.987·83-s − 1.95·85-s − 0.635·89-s + 0.615·95-s + 1.31·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−15.49000762025463, −14.90005595376303, −14.47420182457788, −14.08054369163175, −13.33230042476180, −12.59612692844921, −12.27622862002706, −11.61708402913775, −11.45815423060276, −10.69312011183852, −10.07159580211395, −9.549495085053712, −8.836583615015836, −8.141491460609389, −7.955023263154005, −7.323494928951417, −6.463896809496048, −6.187628460943145, −5.295348951379092, −4.518176394127658, −3.994190385115678, −3.499412333572340, −2.880350653186854, −1.715900389913532, −0.9964744410093838, 0,
0.9964744410093838, 1.715900389913532, 2.880350653186854, 3.499412333572340, 3.994190385115678, 4.518176394127658, 5.295348951379092, 6.187628460943145, 6.463896809496048, 7.323494928951417, 7.955023263154005, 8.141491460609389, 8.836583615015836, 9.549495085053712, 10.07159580211395, 10.69312011183852, 11.45815423060276, 11.61708402913775, 12.27622862002706, 12.59612692844921, 13.33230042476180, 14.08054369163175, 14.47420182457788, 14.90005595376303, 15.49000762025463
