L(s) = 1 | + 2-s − 3-s + 4-s − 1.82·5-s − 6-s + 8-s + 9-s − 1.82·10-s + 4.14·11-s − 12-s + 2.18·13-s + 1.82·15-s + 16-s + 3.35·17-s + 18-s + 7.33·19-s − 1.82·20-s + 4.14·22-s + 23-s − 24-s − 1.65·25-s + 2.18·26-s − 27-s + 5.18·29-s + 1.82·30-s − 10.5·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.817·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.578·10-s + 1.25·11-s − 0.288·12-s + 0.606·13-s + 0.472·15-s + 0.250·16-s + 0.814·17-s + 0.235·18-s + 1.68·19-s − 0.408·20-s + 0.884·22-s + 0.208·23-s − 0.204·24-s − 0.331·25-s + 0.428·26-s − 0.192·27-s + 0.962·29-s + 0.333·30-s − 1.88·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.743498180\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.743498180\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 + 1.82T + 5T^{2} \) |
| 11 | \( 1 - 4.14T + 11T^{2} \) |
| 13 | \( 1 - 2.18T + 13T^{2} \) |
| 17 | \( 1 - 3.35T + 17T^{2} \) |
| 19 | \( 1 - 7.33T + 19T^{2} \) |
| 29 | \( 1 - 5.18T + 29T^{2} \) |
| 31 | \( 1 + 10.5T + 31T^{2} \) |
| 37 | \( 1 + 1.03T + 37T^{2} \) |
| 41 | \( 1 + 7.33T + 41T^{2} \) |
| 43 | \( 1 - 5.36T + 43T^{2} \) |
| 47 | \( 1 + 7.22T + 47T^{2} \) |
| 53 | \( 1 - 4.54T + 53T^{2} \) |
| 59 | \( 1 - 0.471T + 59T^{2} \) |
| 61 | \( 1 - 14.7T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 + 5.18T + 79T^{2} \) |
| 83 | \( 1 + 5.80T + 83T^{2} \) |
| 89 | \( 1 + 6.69T + 89T^{2} \) |
| 97 | \( 1 - 4.47T + 97T^{2} \) |
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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
−7.83450087609910074051639320201, −6.99231196549153479350184727589, −6.71528570387555530226323623341, −5.55258139307146897379593545920, −5.37265541491018826049114210095, −4.22125427862921388386670611142, −3.72133784711388178524495117433, −3.13435109206786038709475851728, −1.67451169162026248380000110977, −0.843739528948761683587663889322,
0.843739528948761683587663889322, 1.67451169162026248380000110977, 3.13435109206786038709475851728, 3.72133784711388178524495117433, 4.22125427862921388386670611142, 5.37265541491018826049114210095, 5.55258139307146897379593545920, 6.71528570387555530226323623341, 6.99231196549153479350184727589, 7.83450087609910074051639320201