| L(s) = 1 | − 2.85·5-s + 4.33·7-s − 3·9-s + 2.15·11-s − 13-s − 1.77·17-s − 0.194·19-s − 1.72·23-s + 3.15·25-s + 10.1·29-s + 4.05·31-s − 12.3·35-s − 9.62·37-s + 8.70·41-s − 5.71·43-s + 8.56·45-s − 6.36·47-s + 11.7·49-s + 9.62·53-s − 6.14·55-s + 10.6·59-s − 1.26·61-s − 13.0·63-s + 2.85·65-s − 67-s − 4.84·71-s − 3.06·73-s + ⋯ |
| L(s) = 1 | − 1.27·5-s + 1.63·7-s − 9-s + 0.649·11-s − 0.277·13-s − 0.430·17-s − 0.0446·19-s − 0.360·23-s + 0.630·25-s + 1.88·29-s + 0.728·31-s − 2.09·35-s − 1.58·37-s + 1.36·41-s − 0.870·43-s + 1.27·45-s − 0.928·47-s + 1.68·49-s + 1.32·53-s − 0.828·55-s + 1.38·59-s − 0.161·61-s − 1.63·63-s + 0.354·65-s − 0.122·67-s − 0.575·71-s − 0.358·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3484 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3484 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.531980950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.531980950\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 + 2.85T + 5T^{2} \) |
| 7 | \( 1 - 4.33T + 7T^{2} \) |
| 11 | \( 1 - 2.15T + 11T^{2} \) |
| 17 | \( 1 + 1.77T + 17T^{2} \) |
| 19 | \( 1 + 0.194T + 19T^{2} \) |
| 23 | \( 1 + 1.72T + 23T^{2} \) |
| 29 | \( 1 - 10.1T + 29T^{2} \) |
| 31 | \( 1 - 4.05T + 31T^{2} \) |
| 37 | \( 1 + 9.62T + 37T^{2} \) |
| 41 | \( 1 - 8.70T + 41T^{2} \) |
| 43 | \( 1 + 5.71T + 43T^{2} \) |
| 47 | \( 1 + 6.36T + 47T^{2} \) |
| 53 | \( 1 - 9.62T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 + 1.26T + 61T^{2} \) |
| 71 | \( 1 + 4.84T + 71T^{2} \) |
| 73 | \( 1 + 3.06T + 73T^{2} \) |
| 79 | \( 1 - 2.54T + 79T^{2} \) |
| 83 | \( 1 - 4.44T + 83T^{2} \) |
| 89 | \( 1 - 7.37T + 89T^{2} \) |
| 97 | \( 1 + 14.0T + 97T^{2} \) |
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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
−8.405072959416871304835644682284, −8.070366767281567397961596064570, −7.23790237777157116660598728123, −6.44985312052851720135215271760, −5.38054919264044039742609687429, −4.66063063678573926165550863027, −4.08513274661913016223491812632, −3.07565347255238536810752238475, −2.00496479466063210369496145338, −0.73565345319349204530994254207,
0.73565345319349204530994254207, 2.00496479466063210369496145338, 3.07565347255238536810752238475, 4.08513274661913016223491812632, 4.66063063678573926165550863027, 5.38054919264044039742609687429, 6.44985312052851720135215271760, 7.23790237777157116660598728123, 8.070366767281567397961596064570, 8.405072959416871304835644682284
