| L(s) = 1 | − 1.25·3-s − 1.33·5-s − 1.42·9-s − 4.71·11-s + 13-s + 1.67·15-s − 7.13·17-s − 0.127·19-s − 5.10·23-s − 3.21·25-s + 5.55·27-s − 0.498·29-s − 1.28·31-s + 5.93·33-s + 0.946·37-s − 1.25·39-s − 1.55·41-s − 8.67·43-s + 1.89·45-s − 0.268·47-s + 8.97·51-s − 5.14·53-s + 6.29·55-s + 0.160·57-s + 10.2·59-s − 4.39·61-s − 1.33·65-s + ⋯ |
| L(s) = 1 | − 0.725·3-s − 0.596·5-s − 0.473·9-s − 1.42·11-s + 0.277·13-s + 0.433·15-s − 1.73·17-s − 0.0293·19-s − 1.06·23-s − 0.643·25-s + 1.06·27-s − 0.0925·29-s − 0.231·31-s + 1.03·33-s + 0.155·37-s − 0.201·39-s − 0.242·41-s − 1.32·43-s + 0.282·45-s − 0.0392·47-s + 1.25·51-s − 0.707·53-s + 0.849·55-s + 0.0213·57-s + 1.33·59-s − 0.562·61-s − 0.165·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2522487844\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2522487844\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 1.25T + 3T^{2} \) |
| 5 | \( 1 + 1.33T + 5T^{2} \) |
| 11 | \( 1 + 4.71T + 11T^{2} \) |
| 17 | \( 1 + 7.13T + 17T^{2} \) |
| 19 | \( 1 + 0.127T + 19T^{2} \) |
| 23 | \( 1 + 5.10T + 23T^{2} \) |
| 29 | \( 1 + 0.498T + 29T^{2} \) |
| 31 | \( 1 + 1.28T + 31T^{2} \) |
| 37 | \( 1 - 0.946T + 37T^{2} \) |
| 41 | \( 1 + 1.55T + 41T^{2} \) |
| 43 | \( 1 + 8.67T + 43T^{2} \) |
| 47 | \( 1 + 0.268T + 47T^{2} \) |
| 53 | \( 1 + 5.14T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 4.39T + 61T^{2} \) |
| 67 | \( 1 - 9.65T + 67T^{2} \) |
| 71 | \( 1 + 0.991T + 71T^{2} \) |
| 73 | \( 1 + 4.01T + 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 + 9.28T + 83T^{2} \) |
| 89 | \( 1 - 15.7T + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−8.274949596362117025593730296262, −7.54266721126612996010968580118, −6.70095792030394427597463048822, −6.03885849908357167645185658394, −5.31947216904659500154837106097, −4.64401433390056541093991200188, −3.83249047474756517656548227241, −2.81378947363132926883476888679, −1.95760445799339669474470836218, −0.26481844993917942091556765135,
0.26481844993917942091556765135, 1.95760445799339669474470836218, 2.81378947363132926883476888679, 3.83249047474756517656548227241, 4.64401433390056541093991200188, 5.31947216904659500154837106097, 6.03885849908357167645185658394, 6.70095792030394427597463048822, 7.54266721126612996010968580118, 8.274949596362117025593730296262
