| L(s) = 1 | + 3-s + 7-s + 9-s + 5·11-s − 13-s + 5·17-s + 21-s + 27-s − 7·29-s + 9·31-s + 5·33-s − 8·37-s − 39-s − 2·41-s − 8·43-s + 9·47-s − 6·49-s + 5·51-s + 11·53-s − 59-s − 7·61-s + 63-s + 15·67-s + 8·71-s + 4·73-s + 5·77-s + 4·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s + 1.50·11-s − 0.277·13-s + 1.21·17-s + 0.218·21-s + 0.192·27-s − 1.29·29-s + 1.61·31-s + 0.870·33-s − 1.31·37-s − 0.160·39-s − 0.312·41-s − 1.21·43-s + 1.31·47-s − 6/7·49-s + 0.700·51-s + 1.51·53-s − 0.130·59-s − 0.896·61-s + 0.125·63-s + 1.83·67-s + 0.949·71-s + 0.468·73-s + 0.569·77-s + 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.584673997\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.584673997\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 15 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 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
−15.98613788252342, −15.15380514272703, −14.97837731231754, −14.28057401780134, −13.91009565178682, −13.41568940132244, −12.54996459097792, −11.95696290182400, −11.78889547762617, −10.88466128173206, −10.26574408640524, −9.593688426042163, −9.261972437314396, −8.436528278142139, −8.086437280163754, −7.268367735889593, −6.788034284173890, −6.076378046737377, −5.269682220093971, −4.655735791788656, −3.678825693218192, −3.511951286151610, −2.371586848157918, −1.625063217779534, −0.8469767292486753,
0.8469767292486753, 1.625063217779534, 2.371586848157918, 3.511951286151610, 3.678825693218192, 4.655735791788656, 5.269682220093971, 6.076378046737377, 6.788034284173890, 7.268367735889593, 8.086437280163754, 8.436528278142139, 9.261972437314396, 9.593688426042163, 10.26574408640524, 10.88466128173206, 11.78889547762617, 11.95696290182400, 12.54996459097792, 13.41568940132244, 13.91009565178682, 14.28057401780134, 14.97837731231754, 15.15380514272703, 15.98613788252342
