| L(s) = 1 | + 1.92·2-s + 2.94·3-s + 1.70·4-s + 1.06·5-s + 5.67·6-s − 3.23·7-s − 0.564·8-s + 5.70·9-s + 2.05·10-s + 1.94·11-s + 5.03·12-s + 5.03·13-s − 6.23·14-s + 3.14·15-s − 4.50·16-s + 2.39·17-s + 10.9·18-s − 1.59·19-s + 1.82·20-s − 9.55·21-s + 3.74·22-s + 1.56·23-s − 1.66·24-s − 3.85·25-s + 9.69·26-s + 7.96·27-s − 5.52·28-s + ⋯ |
| L(s) = 1 | + 1.36·2-s + 1.70·3-s + 0.853·4-s + 0.477·5-s + 2.31·6-s − 1.22·7-s − 0.199·8-s + 1.90·9-s + 0.650·10-s + 0.586·11-s + 1.45·12-s + 1.39·13-s − 1.66·14-s + 0.813·15-s − 1.12·16-s + 0.580·17-s + 2.58·18-s − 0.365·19-s + 0.407·20-s − 2.08·21-s + 0.798·22-s + 0.326·23-s − 0.339·24-s − 0.771·25-s + 1.90·26-s + 1.53·27-s − 1.04·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1369 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1369 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.798831437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.798831437\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 \) |
| good | 2 | \( 1 - 1.92T + 2T^{2} \) |
| 3 | \( 1 - 2.94T + 3T^{2} \) |
| 5 | \( 1 - 1.06T + 5T^{2} \) |
| 7 | \( 1 + 3.23T + 7T^{2} \) |
| 11 | \( 1 - 1.94T + 11T^{2} \) |
| 13 | \( 1 - 5.03T + 13T^{2} \) |
| 17 | \( 1 - 2.39T + 17T^{2} \) |
| 19 | \( 1 + 1.59T + 19T^{2} \) |
| 23 | \( 1 - 1.56T + 23T^{2} \) |
| 29 | \( 1 - 2.17T + 29T^{2} \) |
| 31 | \( 1 + 0.944T + 31T^{2} \) |
| 41 | \( 1 + 9.49T + 41T^{2} \) |
| 43 | \( 1 + 7.59T + 43T^{2} \) |
| 47 | \( 1 + 3.03T + 47T^{2} \) |
| 53 | \( 1 + 8.93T + 53T^{2} \) |
| 59 | \( 1 - 0.235T + 59T^{2} \) |
| 61 | \( 1 - 14.6T + 61T^{2} \) |
| 67 | \( 1 - 9.29T + 67T^{2} \) |
| 71 | \( 1 - 0.841T + 71T^{2} \) |
| 73 | \( 1 + 5.11T + 73T^{2} \) |
| 79 | \( 1 - 6.62T + 79T^{2} \) |
| 83 | \( 1 + 1.99T + 83T^{2} \) |
| 89 | \( 1 - 7.92T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−9.536948610791443588867270576775, −8.796147965292113531598448989538, −8.131675151643865939787262317214, −6.75320282879949568213815270158, −6.40618368414328431500234534153, −5.32403613735928081012505640494, −4.00392776732441349331903316491, −3.52690211818298438878771683038, −2.91738676402918929893210922345, −1.74393152345820537615937311509,
1.74393152345820537615937311509, 2.91738676402918929893210922345, 3.52690211818298438878771683038, 4.00392776732441349331903316491, 5.32403613735928081012505640494, 6.40618368414328431500234534153, 6.75320282879949568213815270158, 8.131675151643865939787262317214, 8.796147965292113531598448989538, 9.536948610791443588867270576775
