| L(s) = 1 | − 5-s + 2·13-s − 6·17-s − 4·19-s + 25-s + 6·29-s + 8·31-s + 6·37-s + 10·41-s − 4·43-s − 8·47-s − 7·49-s + 10·53-s − 12·59-s − 6·61-s − 2·65-s + 4·67-s + 14·73-s − 4·83-s + 6·85-s + 6·89-s + 4·95-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.554·13-s − 1.45·17-s − 0.917·19-s + 1/5·25-s + 1.11·29-s + 1.43·31-s + 0.986·37-s + 1.56·41-s − 0.609·43-s − 1.16·47-s − 49-s + 1.37·53-s − 1.56·59-s − 0.768·61-s − 0.248·65-s + 0.488·67-s + 1.63·73-s − 0.439·83-s + 0.650·85-s + 0.635·89-s + 0.410·95-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.94057276582364, −13.77768982971242, −13.06801367879007, −12.76296546349008, −12.19174164041245, −11.56471356829279, −11.23936641082884, −10.71778019409115, −10.29998201314847, −9.606288899795665, −9.085322767245957, −8.601962428422038, −8.040461671624503, −7.807747512737240, −6.833466184681192, −6.410034215939810, −6.257441962106303, −5.269758533225990, −4.622943320282108, −4.302178707061158, −3.737468671688940, −2.826985741120562, −2.507896526066047, −1.611653991457256, −0.8410750312644827, 0,
0.8410750312644827, 1.611653991457256, 2.507896526066047, 2.826985741120562, 3.737468671688940, 4.302178707061158, 4.622943320282108, 5.269758533225990, 6.257441962106303, 6.410034215939810, 6.833466184681192, 7.807747512737240, 8.040461671624503, 8.601962428422038, 9.085322767245957, 9.606288899795665, 10.29998201314847, 10.71778019409115, 11.23936641082884, 11.56471356829279, 12.19174164041245, 12.76296546349008, 13.06801367879007, 13.77768982971242, 13.94057276582364
