| L(s) = 1 | − 3·3-s + 5·5-s + 9·9-s − 15·15-s + 14·17-s + 22·19-s − 34·23-s + 25·25-s − 27·27-s + 2·31-s + 45·45-s + 14·47-s + 49·49-s − 42·51-s − 86·53-s − 66·57-s + 118·61-s + 102·69-s − 75·75-s + 98·79-s + 81·81-s + 154·83-s + 70·85-s − 6·93-s + 110·95-s + 106·107-s + 22·109-s + ⋯ |
| L(s) = 1 | − 3-s + 5-s + 9-s − 15-s + 0.823·17-s + 1.15·19-s − 1.47·23-s + 25-s − 27-s + 2/31·31-s + 45-s + 0.297·47-s + 49-s − 0.823·51-s − 1.62·53-s − 1.15·57-s + 1.93·61-s + 1.47·69-s − 75-s + 1.24·79-s + 81-s + 1.85·83-s + 0.823·85-s − 0.0645·93-s + 1.15·95-s + 0.990·107-s + 0.201·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.710789635\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.710789635\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| good | 7 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( ( 1 - p T )( 1 + p T ) \) |
| 17 | \( 1 - 14 T + p^{2} T^{2} \) |
| 19 | \( 1 - 22 T + p^{2} T^{2} \) |
| 23 | \( 1 + 34 T + p^{2} T^{2} \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( 1 - 2 T + p^{2} T^{2} \) |
| 37 | \( ( 1 - p T )( 1 + p T ) \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( 1 - 14 T + p^{2} T^{2} \) |
| 53 | \( 1 + 86 T + p^{2} T^{2} \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 - 118 T + p^{2} T^{2} \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( ( 1 - p T )( 1 + p T ) \) |
| 79 | \( 1 - 98 T + p^{2} T^{2} \) |
| 83 | \( 1 - 154 T + p^{2} T^{2} \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( ( 1 - p T )( 1 + p T ) \) |
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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
−9.925668948784454626557899350945, −9.335852816491659953378042539387, −8.038772993624185170257372302308, −7.14805377170756364569631604980, −6.18640057988757255764426764159, −5.60659555763739050823007864989, −4.81659415479777361561111306117, −3.54846471045103199402469838026, −2.06435819921759419808522984177, −0.897132195309456049095487338102,
0.897132195309456049095487338102, 2.06435819921759419808522984177, 3.54846471045103199402469838026, 4.81659415479777361561111306117, 5.60659555763739050823007864989, 6.18640057988757255764426764159, 7.14805377170756364569631604980, 8.038772993624185170257372302308, 9.335852816491659953378042539387, 9.925668948784454626557899350945
