| L(s) = 1 | − 3-s − 5-s + 2·7-s + 9-s + 2·11-s + 15-s − 7·17-s + 6·19-s − 2·21-s + 6·23-s − 4·25-s − 27-s + 29-s + 4·31-s − 2·33-s − 2·35-s + 37-s − 9·41-s + 6·43-s − 45-s + 6·47-s − 3·49-s + 7·51-s + 9·53-s − 2·55-s − 6·57-s − 61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.603·11-s + 0.258·15-s − 1.69·17-s + 1.37·19-s − 0.436·21-s + 1.25·23-s − 4/5·25-s − 0.192·27-s + 0.185·29-s + 0.718·31-s − 0.348·33-s − 0.338·35-s + 0.164·37-s − 1.40·41-s + 0.914·43-s − 0.149·45-s + 0.875·47-s − 3/7·49-s + 0.980·51-s + 1.23·53-s − 0.269·55-s − 0.794·57-s − 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.895856629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.895856629\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.21357569835520, −14.57178171651977, −13.92456819692987, −13.42736248955985, −13.05252748445199, −12.07036995644291, −11.81883385549138, −11.47414118578648, −10.81284359998324, −10.48108486094552, −9.562014720745120, −9.141716835023871, −8.576278959950457, −7.884126013778847, −7.373453126635141, −6.742184161336370, −6.340532072249114, −5.391795764530273, −5.032531885485926, −4.331687334675099, −3.865045163186278, −2.974792702584524, −2.151616905755663, −1.317756781195863, −0.5877870892332736,
0.5877870892332736, 1.317756781195863, 2.151616905755663, 2.974792702584524, 3.865045163186278, 4.331687334675099, 5.032531885485926, 5.391795764530273, 6.340532072249114, 6.742184161336370, 7.373453126635141, 7.884126013778847, 8.576278959950457, 9.141716835023871, 9.562014720745120, 10.48108486094552, 10.81284359998324, 11.47414118578648, 11.81883385549138, 12.07036995644291, 13.05252748445199, 13.42736248955985, 13.92456819692987, 14.57178171651977, 15.21357569835520
