L(s) = 1 | − 2·4-s − 4·7-s + 5·13-s + 4·16-s + 8·19-s + 8·28-s + 11·31-s − 37-s − 13·43-s + 9·49-s − 10·52-s + 14·61-s − 8·64-s + 5·67-s + 17·73-s − 16·76-s − 13·79-s − 20·91-s + 14·97-s − 7·103-s + 17·109-s − 16·112-s + ⋯ |
L(s) = 1 | − 4-s − 1.51·7-s + 1.38·13-s + 16-s + 1.83·19-s + 1.51·28-s + 1.97·31-s − 0.164·37-s − 1.98·43-s + 9/7·49-s − 1.38·52-s + 1.79·61-s − 64-s + 0.610·67-s + 1.98·73-s − 1.83·76-s − 1.46·79-s − 2.09·91-s + 1.42·97-s − 0.689·103-s + 1.62·109-s − 1.51·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.033136608\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.033136608\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 11 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 17 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−10.04820957110891225559888247416, −9.804478021252617404784341015739, −8.831795318502833330339208828788, −8.103539044020415994829049592764, −6.83497060476461397094513639636, −6.01724090331191530541613327241, −5.05243608272378186060105532251, −3.74647317217248951569851723821, −3.13936473638444884836235755024, −0.895025331879932514186462794519,
0.895025331879932514186462794519, 3.13936473638444884836235755024, 3.74647317217248951569851723821, 5.05243608272378186060105532251, 6.01724090331191530541613327241, 6.83497060476461397094513639636, 8.103539044020415994829049592764, 8.831795318502833330339208828788, 9.804478021252617404784341015739, 10.04820957110891225559888247416