| L(s) = 1 | + 3·5-s + 7-s + 2·11-s + 13-s + 6·17-s − 2·19-s − 23-s + 4·25-s − 3·29-s − 8·31-s + 3·35-s − 7·37-s − 9·41-s − 43-s + 7·47-s + 49-s + 6·53-s + 6·55-s + 2·61-s + 3·65-s − 4·67-s + 10·71-s − 14·73-s + 2·77-s − 12·79-s + 4·83-s + 18·85-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.377·7-s + 0.603·11-s + 0.277·13-s + 1.45·17-s − 0.458·19-s − 0.208·23-s + 4/5·25-s − 0.557·29-s − 1.43·31-s + 0.507·35-s − 1.15·37-s − 1.40·41-s − 0.152·43-s + 1.02·47-s + 1/7·49-s + 0.824·53-s + 0.809·55-s + 0.256·61-s + 0.372·65-s − 0.488·67-s + 1.18·71-s − 1.63·73-s + 0.227·77-s − 1.35·79-s + 0.439·83-s + 1.95·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−14.09037411831865, −13.63635542388078, −13.18783971506112, −12.66820302341421, −12.03834524955371, −11.79584279825310, −11.00064053850122, −10.52222739024704, −10.15418132329474, −9.621654832826611, −9.131254082313890, −8.692033381180848, −8.146616448015802, −7.422125029724411, −6.990515983945161, −6.379377575076159, −5.772620760687939, −5.437884935567986, −5.030911519422253, −4.025361413256973, −3.684997430597855, −2.917030154310522, −2.160807385264113, −1.590177961463199, −1.212027151909718, 0,
1.212027151909718, 1.590177961463199, 2.160807385264113, 2.917030154310522, 3.684997430597855, 4.025361413256973, 5.030911519422253, 5.437884935567986, 5.772620760687939, 6.379377575076159, 6.990515983945161, 7.422125029724411, 8.146616448015802, 8.692033381180848, 9.131254082313890, 9.621654832826611, 10.15418132329474, 10.52222739024704, 11.00064053850122, 11.79584279825310, 12.03834524955371, 12.66820302341421, 13.18783971506112, 13.63635542388078, 14.09037411831865
