L(s) = 1 | − 8·4-s − 17·7-s + 70·13-s + 64·16-s + 107·19-s + 136·28-s − 289·31-s − 323·37-s − 71·43-s − 54·49-s − 560·52-s − 901·61-s − 512·64-s + 880·67-s + 919·73-s − 856·76-s − 1.38e3·79-s − 1.19e3·91-s − 1.85e3·97-s + 1.80e3·103-s − 1.56e3·109-s − 1.08e3·112-s + ⋯ |
L(s) = 1 | − 4-s − 0.917·7-s + 1.49·13-s + 16-s + 1.29·19-s + 0.917·28-s − 1.67·31-s − 1.43·37-s − 0.251·43-s − 0.157·49-s − 1.49·52-s − 1.89·61-s − 64-s + 1.60·67-s + 1.47·73-s − 1.29·76-s − 1.97·79-s − 1.37·91-s − 1.93·97-s + 1.72·103-s − 1.37·109-s − 0.917·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + p^{3} T^{2} \) |
| 7 | \( 1 + 17 T + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 - 70 T + p^{3} T^{2} \) |
| 17 | \( 1 + p^{3} T^{2} \) |
| 19 | \( 1 - 107 T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + p^{3} T^{2} \) |
| 31 | \( 1 + 289 T + p^{3} T^{2} \) |
| 37 | \( 1 + 323 T + p^{3} T^{2} \) |
| 41 | \( 1 + p^{3} T^{2} \) |
| 43 | \( 1 + 71 T + p^{3} T^{2} \) |
| 47 | \( 1 + p^{3} T^{2} \) |
| 53 | \( 1 + p^{3} T^{2} \) |
| 59 | \( 1 + p^{3} T^{2} \) |
| 61 | \( 1 + 901 T + p^{3} T^{2} \) |
| 67 | \( 1 - 880 T + p^{3} T^{2} \) |
| 71 | \( 1 + p^{3} T^{2} \) |
| 73 | \( 1 - 919 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1387 T + p^{3} T^{2} \) |
| 83 | \( 1 + p^{3} T^{2} \) |
| 89 | \( 1 + p^{3} T^{2} \) |
| 97 | \( 1 + 1853 T + p^{3} T^{2} \) |
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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
−9.511014017572136753001734678425, −8.967529609515399255111202323022, −8.081324869396578531683138568182, −6.99803821873125080443977344778, −5.93712079797926490478855752544, −5.16956989363084313614834467004, −3.82426391502226617479874054683, −3.27247442286714994876927027987, −1.33231959963257179621788545111, 0,
1.33231959963257179621788545111, 3.27247442286714994876927027987, 3.82426391502226617479874054683, 5.16956989363084313614834467004, 5.93712079797926490478855752544, 6.99803821873125080443977344778, 8.081324869396578531683138568182, 8.967529609515399255111202323022, 9.511014017572136753001734678425