| L(s) = 1 | + 3-s − 3·5-s + 9-s − 5·11-s + 13-s − 3·15-s + 17-s − 3·19-s + 5·23-s + 4·25-s + 27-s + 2·29-s − 5·33-s − 2·37-s + 39-s − 7·41-s + 3·43-s − 3·45-s + 12·47-s + 51-s − 8·53-s + 15·55-s − 3·57-s + 8·59-s + 6·61-s − 3·65-s − 8·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s + 1/3·9-s − 1.50·11-s + 0.277·13-s − 0.774·15-s + 0.242·17-s − 0.688·19-s + 1.04·23-s + 4/5·25-s + 0.192·27-s + 0.371·29-s − 0.870·33-s − 0.328·37-s + 0.160·39-s − 1.09·41-s + 0.457·43-s − 0.447·45-s + 1.75·47-s + 0.140·51-s − 1.09·53-s + 2.02·55-s − 0.397·57-s + 1.04·59-s + 0.768·61-s − 0.372·65-s − 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−13.40921207167267, −13.06987481883692, −12.55964648492209, −12.19653596047616, −11.62403266090465, −11.03752734095400, −10.75638098303585, −10.23352061776589, −9.765410033695077, −8.991557934064057, −8.543305367935839, −8.273444797578560, −7.744914158826169, −7.270444672293747, −6.962500062321218, −6.210702006982038, −5.445881910309813, −5.096996364630536, −4.322788218937490, −4.077687684505993, −3.305199814015803, −2.909998608932745, −2.393194088282936, −1.544933882189503, −0.6977128699484564, 0,
0.6977128699484564, 1.544933882189503, 2.393194088282936, 2.909998608932745, 3.305199814015803, 4.077687684505993, 4.322788218937490, 5.096996364630536, 5.445881910309813, 6.210702006982038, 6.962500062321218, 7.270444672293747, 7.744914158826169, 8.273444797578560, 8.543305367935839, 8.991557934064057, 9.765410033695077, 10.23352061776589, 10.75638098303585, 11.03752734095400, 11.62403266090465, 12.19653596047616, 12.55964648492209, 13.06987481883692, 13.40921207167267
