| L(s) = 1 | + 3.77·5-s − 4.71·7-s + 5.30·11-s + 3.30·13-s + 1.05·17-s + 0.837·19-s + 9.27·25-s + 3.75·29-s − 31-s − 17.8·35-s + 4.24·37-s − 7.21·41-s − 0.809·43-s + 5.55·47-s + 15.2·49-s − 1.42·53-s + 20.0·55-s − 3.89·59-s − 13.6·61-s + 12.4·65-s − 15.1·67-s + 4.83·71-s − 9.11·73-s − 25.0·77-s + 9.93·79-s + 5.80·83-s + 4·85-s + ⋯ |
| L(s) = 1 | + 1.68·5-s − 1.78·7-s + 1.60·11-s + 0.917·13-s + 0.256·17-s + 0.192·19-s + 1.85·25-s + 0.696·29-s − 0.179·31-s − 3.01·35-s + 0.698·37-s − 1.12·41-s − 0.123·43-s + 0.810·47-s + 2.18·49-s − 0.195·53-s + 2.70·55-s − 0.507·59-s − 1.75·61-s + 1.55·65-s − 1.84·67-s + 0.574·71-s − 1.06·73-s − 2.85·77-s + 1.11·79-s + 0.637·83-s + 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.655332684\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.655332684\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 5 | \( 1 - 3.77T + 5T^{2} \) |
| 7 | \( 1 + 4.71T + 7T^{2} \) |
| 11 | \( 1 - 5.30T + 11T^{2} \) |
| 13 | \( 1 - 3.30T + 13T^{2} \) |
| 17 | \( 1 - 1.05T + 17T^{2} \) |
| 19 | \( 1 - 0.837T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 3.75T + 29T^{2} \) |
| 37 | \( 1 - 4.24T + 37T^{2} \) |
| 41 | \( 1 + 7.21T + 41T^{2} \) |
| 43 | \( 1 + 0.809T + 43T^{2} \) |
| 47 | \( 1 - 5.55T + 47T^{2} \) |
| 53 | \( 1 + 1.42T + 53T^{2} \) |
| 59 | \( 1 + 3.89T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 + 15.1T + 67T^{2} \) |
| 71 | \( 1 - 4.83T + 71T^{2} \) |
| 73 | \( 1 + 9.11T + 73T^{2} \) |
| 79 | \( 1 - 9.93T + 79T^{2} \) |
| 83 | \( 1 - 5.80T + 83T^{2} \) |
| 89 | \( 1 - 9.11T + 89T^{2} \) |
| 97 | \( 1 + 1.33T + 97T^{2} \) |
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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
−8.884069260539594987088877946334, −7.39052405686087236299135002635, −6.50197661701866625392430313338, −6.24016091075552906674468704150, −5.81035733884210768108760363306, −4.62546081342297595357289259781, −3.55552989567525059086846467710, −3.01357347062244040463556646080, −1.86984959854100982667035162467, −0.964506117885466793161119200401,
0.964506117885466793161119200401, 1.86984959854100982667035162467, 3.01357347062244040463556646080, 3.55552989567525059086846467710, 4.62546081342297595357289259781, 5.81035733884210768108760363306, 6.24016091075552906674468704150, 6.50197661701866625392430313338, 7.39052405686087236299135002635, 8.884069260539594987088877946334
