| L(s) = 1 | − 4·5-s + 4·7-s + 6·17-s − 19-s + 6·23-s + 11·25-s + 6·29-s − 10·31-s − 16·35-s − 12·37-s − 10·41-s + 4·43-s − 6·47-s + 9·49-s − 2·53-s − 12·59-s + 6·61-s + 8·67-s − 10·73-s − 10·79-s + 12·83-s − 24·85-s − 6·89-s + 4·95-s + 6·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 1.51·7-s + 1.45·17-s − 0.229·19-s + 1.25·23-s + 11/5·25-s + 1.11·29-s − 1.79·31-s − 2.70·35-s − 1.97·37-s − 1.56·41-s + 0.609·43-s − 0.875·47-s + 9/7·49-s − 0.274·53-s − 1.56·59-s + 0.768·61-s + 0.977·67-s − 1.17·73-s − 1.12·79-s + 1.31·83-s − 2.60·85-s − 0.635·89-s + 0.410·95-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| 113 | \( 1 + T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−12.72765300890173, −12.17611527371984, −12.02719268938009, −11.56920752406107, −11.08366955017755, −10.72877574242144, −10.42965943231499, −9.677945280099725, −8.989888685796550, −8.551679073945658, −8.286464731871224, −7.798056140372863, −7.421964237633995, −7.010489036240527, −6.559781393495338, −5.559913454075532, −5.129554626192959, −4.931236604461312, −4.243866307548979, −3.814037512741049, −3.199022094439632, −2.955929612712226, −1.799332453492456, −1.459327811169159, −0.7479864310857971, 0,
0.7479864310857971, 1.459327811169159, 1.799332453492456, 2.955929612712226, 3.199022094439632, 3.814037512741049, 4.243866307548979, 4.931236604461312, 5.129554626192959, 5.559913454075532, 6.559781393495338, 7.010489036240527, 7.421964237633995, 7.798056140372863, 8.286464731871224, 8.551679073945658, 8.989888685796550, 9.677945280099725, 10.42965943231499, 10.72877574242144, 11.08366955017755, 11.56920752406107, 12.02719268938009, 12.17611527371984, 12.72765300890173
