| L(s) = 1 | − 5-s + 4.42·7-s + 11-s − 0.622·13-s + 5.18·17-s − 7.05·19-s + 8.85·23-s + 25-s + 7.80·29-s − 2.75·31-s − 4.42·35-s − 2·37-s + 0.193·41-s − 5.67·43-s − 2.75·47-s + 12.6·49-s + 10.8·53-s − 55-s − 4.85·59-s + 6.85·61-s + 0.622·65-s + 1.24·67-s + 2.75·71-s + 4.23·73-s + 4.42·77-s − 8.56·79-s + 0.133·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.67·7-s + 0.301·11-s − 0.172·13-s + 1.25·17-s − 1.61·19-s + 1.84·23-s + 0.200·25-s + 1.44·29-s − 0.494·31-s − 0.748·35-s − 0.328·37-s + 0.0302·41-s − 0.865·43-s − 0.401·47-s + 1.80·49-s + 1.49·53-s − 0.134·55-s − 0.632·59-s + 0.877·61-s + 0.0771·65-s + 0.152·67-s + 0.327·71-s + 0.495·73-s + 0.504·77-s − 0.963·79-s + 0.0146·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.585663210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.585663210\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 - 4.42T + 7T^{2} \) |
| 13 | \( 1 + 0.622T + 13T^{2} \) |
| 17 | \( 1 - 5.18T + 17T^{2} \) |
| 19 | \( 1 + 7.05T + 19T^{2} \) |
| 23 | \( 1 - 8.85T + 23T^{2} \) |
| 29 | \( 1 - 7.80T + 29T^{2} \) |
| 31 | \( 1 + 2.75T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 0.193T + 41T^{2} \) |
| 43 | \( 1 + 5.67T + 43T^{2} \) |
| 47 | \( 1 + 2.75T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 + 4.85T + 59T^{2} \) |
| 61 | \( 1 - 6.85T + 61T^{2} \) |
| 67 | \( 1 - 1.24T + 67T^{2} \) |
| 71 | \( 1 - 2.75T + 71T^{2} \) |
| 73 | \( 1 - 4.23T + 73T^{2} \) |
| 79 | \( 1 + 8.56T + 79T^{2} \) |
| 83 | \( 1 - 0.133T + 83T^{2} \) |
| 89 | \( 1 + 5.61T + 89T^{2} \) |
| 97 | \( 1 - 7.24T + 97T^{2} \) |
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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
−7.931356002241525930031641487621, −7.19665138114311154618078558306, −6.62810016614733550595118601852, −5.57772556426549948982824224944, −4.94489732808917889185902203045, −4.44620091244564769259023248606, −3.59143215059113699969218219347, −2.63910717217144601011097778364, −1.66577982024650155113017982236, −0.856538682856472707899083968668,
0.856538682856472707899083968668, 1.66577982024650155113017982236, 2.63910717217144601011097778364, 3.59143215059113699969218219347, 4.44620091244564769259023248606, 4.94489732808917889185902203045, 5.57772556426549948982824224944, 6.62810016614733550595118601852, 7.19665138114311154618078558306, 7.931356002241525930031641487621
