L(s) = 1 | + 89·13-s − 163·19-s − 125·25-s − 19·31-s − 433·37-s + 449·43-s + 182·61-s + 1.00e3·67-s − 919·73-s + 503·79-s − 1.33e3·97-s − 19·103-s − 1.56e3·109-s + ⋯ |
L(s) = 1 | + 1.89·13-s − 1.96·19-s − 25-s − 0.110·31-s − 1.92·37-s + 1.59·43-s + 0.382·61-s + 1.83·67-s − 1.47·73-s + 0.716·79-s − 1.39·97-s − 0.0181·103-s − 1.37·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 - 89 T + p^{3} T^{2} \) |
| 17 | \( 1 + p^{3} T^{2} \) |
| 19 | \( 1 + 163 T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + p^{3} T^{2} \) |
| 31 | \( 1 + 19 T + p^{3} T^{2} \) |
| 37 | \( 1 + 433 T + p^{3} T^{2} \) |
| 41 | \( 1 + p^{3} T^{2} \) |
| 43 | \( 1 - 449 T + p^{3} T^{2} \) |
| 47 | \( 1 + p^{3} T^{2} \) |
| 53 | \( 1 + p^{3} T^{2} \) |
| 59 | \( 1 + p^{3} T^{2} \) |
| 61 | \( 1 - 182 T + p^{3} T^{2} \) |
| 67 | \( 1 - 1007 T + p^{3} T^{2} \) |
| 71 | \( 1 + p^{3} T^{2} \) |
| 73 | \( 1 + 919 T + p^{3} T^{2} \) |
| 79 | \( 1 - 503 T + p^{3} T^{2} \) |
| 83 | \( 1 + p^{3} T^{2} \) |
| 89 | \( 1 + p^{3} T^{2} \) |
| 97 | \( 1 + 1330 T + p^{3} T^{2} \) |
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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
−8.584050236184648241912289015518, −7.923691641783472199781348495558, −6.78672909301721518149033887856, −6.18720721635831425964345185466, −5.40816605755441490751015948002, −4.14984695869246579901456545077, −3.67019387035805425678120614093, −2.32800845535865852702276396112, −1.34516728263619274132007212632, 0,
1.34516728263619274132007212632, 2.32800845535865852702276396112, 3.67019387035805425678120614093, 4.14984695869246579901456545077, 5.40816605755441490751015948002, 6.18720721635831425964345185466, 6.78672909301721518149033887856, 7.923691641783472199781348495558, 8.584050236184648241912289015518