| L(s) = 1 | + 5-s + 11-s + 7·17-s − 19-s − 2·23-s + 25-s + 8·29-s − 2·31-s − 9·37-s + 41-s − 9·43-s − 7·47-s − 7·49-s + 12·53-s + 55-s − 10·59-s − 2·61-s − 13·67-s − 6·71-s − 4·73-s − 2·79-s + 6·83-s + 7·85-s + 10·89-s − 95-s + 5·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.301·11-s + 1.69·17-s − 0.229·19-s − 0.417·23-s + 1/5·25-s + 1.48·29-s − 0.359·31-s − 1.47·37-s + 0.156·41-s − 1.37·43-s − 1.02·47-s − 49-s + 1.64·53-s + 0.134·55-s − 1.30·59-s − 0.256·61-s − 1.58·67-s − 0.712·71-s − 0.468·73-s − 0.225·79-s + 0.658·83-s + 0.759·85-s + 1.05·89-s − 0.102·95-s + 0.507·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−12.72800769575757, −12.28314959063575, −11.92883921864127, −11.66347118037703, −10.90139204938401, −10.37843949053518, −10.12685468368767, −9.795530933356546, −9.097029508823281, −8.746405661931504, −8.235365586456490, −7.768793312948981, −7.286759515700646, −6.718149603595705, −6.258829461836378, −5.875092112494886, −5.204729196721732, −4.936599740493782, −4.288106946765974, −3.627902377227528, −3.151735807861499, −2.787639499233649, −1.779678861417814, −1.592772925163211, −0.8343672293370261, 0,
0.8343672293370261, 1.592772925163211, 1.779678861417814, 2.787639499233649, 3.151735807861499, 3.627902377227528, 4.288106946765974, 4.936599740493782, 5.204729196721732, 5.875092112494886, 6.258829461836378, 6.718149603595705, 7.286759515700646, 7.768793312948981, 8.235365586456490, 8.746405661931504, 9.097029508823281, 9.795530933356546, 10.12685468368767, 10.37843949053518, 10.90139204938401, 11.66347118037703, 11.92883921864127, 12.28314959063575, 12.72800769575757
