| L(s) = 1 | + 2-s + 3-s + 4-s + 2.11·5-s + 6-s + 1.85·7-s + 8-s + 9-s + 2.11·10-s − 0.995·11-s + 12-s + 4.79·13-s + 1.85·14-s + 2.11·15-s + 16-s − 17-s + 18-s + 4.98·19-s + 2.11·20-s + 1.85·21-s − 0.995·22-s − 0.704·23-s + 24-s − 0.513·25-s + 4.79·26-s + 27-s + 1.85·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.947·5-s + 0.408·6-s + 0.700·7-s + 0.353·8-s + 0.333·9-s + 0.669·10-s − 0.300·11-s + 0.288·12-s + 1.33·13-s + 0.495·14-s + 0.546·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s + 1.14·19-s + 0.473·20-s + 0.404·21-s − 0.212·22-s − 0.146·23-s + 0.204·24-s − 0.102·25-s + 0.940·26-s + 0.192·27-s + 0.350·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.782092256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.782092256\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 5 | \( 1 - 2.11T + 5T^{2} \) |
| 7 | \( 1 - 1.85T + 7T^{2} \) |
| 11 | \( 1 + 0.995T + 11T^{2} \) |
| 13 | \( 1 - 4.79T + 13T^{2} \) |
| 19 | \( 1 - 4.98T + 19T^{2} \) |
| 23 | \( 1 + 0.704T + 23T^{2} \) |
| 29 | \( 1 + 3.89T + 29T^{2} \) |
| 31 | \( 1 + 8.75T + 31T^{2} \) |
| 37 | \( 1 - 6.77T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 2.56T + 43T^{2} \) |
| 47 | \( 1 + 1.28T + 47T^{2} \) |
| 53 | \( 1 - 4.45T + 53T^{2} \) |
| 61 | \( 1 + 6.40T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 + 0.787T + 71T^{2} \) |
| 73 | \( 1 + 6.86T + 73T^{2} \) |
| 79 | \( 1 - 1.42T + 79T^{2} \) |
| 83 | \( 1 - 8.93T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 + 2.30T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.78508778035645950919614530044, −7.58326392333866691645345868641, −6.47024127107150345048350795498, −5.78903665435484993743383187614, −5.35189833098639912711970123706, −4.36059066244446643333783623162, −3.67796196139117932215778583843, −2.80064574650258345942554482722, −1.94492166898634058371129410217, −1.25338977569637865823051522351,
1.25338977569637865823051522351, 1.94492166898634058371129410217, 2.80064574650258345942554482722, 3.67796196139117932215778583843, 4.36059066244446643333783623162, 5.35189833098639912711970123706, 5.78903665435484993743383187614, 6.47024127107150345048350795498, 7.58326392333866691645345868641, 7.78508778035645950919614530044
