L(s) = 1 | − 3-s − 2·5-s + 7-s + 9-s − 6·11-s + 2·15-s − 17-s − 2·19-s − 21-s − 25-s − 27-s + 4·29-s + 6·33-s − 2·35-s − 8·37-s + 2·41-s − 4·43-s − 2·45-s + 8·47-s + 49-s + 51-s + 14·53-s + 12·55-s + 2·57-s − 6·59-s + 10·61-s + 63-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.80·11-s + 0.516·15-s − 0.242·17-s − 0.458·19-s − 0.218·21-s − 1/5·25-s − 0.192·27-s + 0.742·29-s + 1.04·33-s − 0.338·35-s − 1.31·37-s + 0.312·41-s − 0.609·43-s − 0.298·45-s + 1.16·47-s + 1/7·49-s + 0.140·51-s + 1.92·53-s + 1.61·55-s + 0.264·57-s − 0.781·59-s + 1.28·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−15.74759435451892, −15.32136375334770, −14.98747734225921, −13.91242094333972, −13.72194667002657, −12.85348633093185, −12.50228900849046, −11.96724059417352, −11.31112213174999, −10.93343976606025, −10.31636411075803, −10.00125943156744, −8.995037109807529, −8.222518101997097, −8.132252828541716, −7.214817951072803, −6.962126433019163, −5.960056911740168, −5.373434942482353, −4.899395757448349, −4.207434025770579, −3.577732166508391, −2.640716481780136, −2.028332969360980, −0.7850663140986856, 0,
0.7850663140986856, 2.028332969360980, 2.640716481780136, 3.577732166508391, 4.207434025770579, 4.899395757448349, 5.373434942482353, 5.960056911740168, 6.962126433019163, 7.214817951072803, 8.132252828541716, 8.222518101997097, 8.995037109807529, 10.00125943156744, 10.31636411075803, 10.93343976606025, 11.31112213174999, 11.96724059417352, 12.50228900849046, 12.85348633093185, 13.72194667002657, 13.91242094333972, 14.98747734225921, 15.32136375334770, 15.74759435451892