| L(s) = 1 | + 5-s + 2·7-s + 13-s − 5·17-s + 6·19-s − 2·23-s − 4·25-s − 9·29-s − 2·31-s + 2·35-s + 3·37-s − 5·41-s − 2·47-s − 3·49-s + 9·53-s + 8·59-s + 6·61-s + 65-s − 2·67-s − 12·71-s + 2·73-s + 10·79-s − 6·83-s − 5·85-s + 9·89-s + 2·91-s + 6·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s + 0.277·13-s − 1.21·17-s + 1.37·19-s − 0.417·23-s − 4/5·25-s − 1.67·29-s − 0.359·31-s + 0.338·35-s + 0.493·37-s − 0.780·41-s − 0.291·47-s − 3/7·49-s + 1.23·53-s + 1.04·59-s + 0.768·61-s + 0.124·65-s − 0.244·67-s − 1.42·71-s + 0.234·73-s + 1.12·79-s − 0.658·83-s − 0.542·85-s + 0.953·89-s + 0.209·91-s + 0.615·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−14.39909335358654, −13.78543403797655, −13.39391221585581, −13.14223539817168, −12.37195286441779, −11.68384900486178, −11.42743301166704, −11.03872557540944, −10.28583711182322, −9.858458377439730, −9.279385817350898, −8.875040963271441, −8.220003626776572, −7.733579904751183, −7.190713916352310, −6.669404506445241, −5.892995692684043, −5.519879956919244, −4.999974046625444, −4.267619225922946, −3.762818100195758, −3.075095627830568, −2.112484788321973, −1.881321819480286, −1.008368121121711, 0,
1.008368121121711, 1.881321819480286, 2.112484788321973, 3.075095627830568, 3.762818100195758, 4.267619225922946, 4.999974046625444, 5.519879956919244, 5.892995692684043, 6.669404506445241, 7.190713916352310, 7.733579904751183, 8.220003626776572, 8.875040963271441, 9.279385817350898, 9.858458377439730, 10.28583711182322, 11.03872557540944, 11.42743301166704, 11.68384900486178, 12.37195286441779, 13.14223539817168, 13.39391221585581, 13.78543403797655, 14.39909335358654
