| L(s) = 1 | − 4·7-s − 6·11-s − 13-s − 4·17-s − 2·19-s + 6·23-s + 10·29-s − 4·31-s + 6·37-s − 10·41-s − 8·47-s + 9·49-s − 6·53-s − 6·59-s − 6·61-s − 12·67-s − 2·73-s + 24·77-s + 8·79-s + 4·83-s − 14·89-s + 4·91-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 1.80·11-s − 0.277·13-s − 0.970·17-s − 0.458·19-s + 1.25·23-s + 1.85·29-s − 0.718·31-s + 0.986·37-s − 1.56·41-s − 1.16·47-s + 9/7·49-s − 0.824·53-s − 0.781·59-s − 0.768·61-s − 1.46·67-s − 0.234·73-s + 2.73·77-s + 0.900·79-s + 0.439·83-s − 1.48·89-s + 0.419·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.31322112888430, −14.79094245575078, −13.76398680513850, −13.57382900576152, −12.94496991415388, −12.74362258577808, −12.23183028986485, −11.43548227348815, −10.78583151524352, −10.50145758976066, −9.901409702054035, −9.479068730604430, −8.793268926656659, −8.346259735741592, −7.651148501456474, −7.112620246223501, −6.389123622827339, −6.289213676544470, −5.220678592724634, −4.922082905291370, −4.214883405140946, −3.262717440703718, −2.849444704955776, −2.448256005988272, −1.341873386591828, 0, 0,
1.341873386591828, 2.448256005988272, 2.849444704955776, 3.262717440703718, 4.214883405140946, 4.922082905291370, 5.220678592724634, 6.289213676544470, 6.389123622827339, 7.112620246223501, 7.651148501456474, 8.346259735741592, 8.793268926656659, 9.479068730604430, 9.901409702054035, 10.50145758976066, 10.78583151524352, 11.43548227348815, 12.23183028986485, 12.74362258577808, 12.94496991415388, 13.57382900576152, 13.76398680513850, 14.79094245575078, 15.31322112888430
