| L(s) = 1 | + 2.30·2-s − 2.54·3-s + 3.32·4-s − 2.49·5-s − 5.86·6-s + 1.60·7-s + 3.05·8-s + 3.46·9-s − 5.76·10-s + 0.733·11-s − 8.45·12-s − 1.89·13-s + 3.69·14-s + 6.35·15-s + 0.404·16-s − 4.37·17-s + 7.99·18-s − 4.63·19-s − 8.30·20-s − 4.07·21-s + 1.69·22-s − 7.11·23-s − 7.77·24-s + 1.24·25-s − 4.37·26-s − 1.18·27-s + 5.32·28-s + ⋯ |
| L(s) = 1 | + 1.63·2-s − 1.46·3-s + 1.66·4-s − 1.11·5-s − 2.39·6-s + 0.605·7-s + 1.08·8-s + 1.15·9-s − 1.82·10-s + 0.221·11-s − 2.44·12-s − 0.525·13-s + 0.987·14-s + 1.64·15-s + 0.101·16-s − 1.06·17-s + 1.88·18-s − 1.06·19-s − 1.85·20-s − 0.888·21-s + 0.360·22-s − 1.48·23-s − 1.58·24-s + 0.248·25-s − 0.858·26-s − 0.227·27-s + 1.00·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 31 | \( 1 \) |
| good | 2 | \( 1 - 2.30T + 2T^{2} \) |
| 3 | \( 1 + 2.54T + 3T^{2} \) |
| 5 | \( 1 + 2.49T + 5T^{2} \) |
| 7 | \( 1 - 1.60T + 7T^{2} \) |
| 11 | \( 1 - 0.733T + 11T^{2} \) |
| 13 | \( 1 + 1.89T + 13T^{2} \) |
| 17 | \( 1 + 4.37T + 17T^{2} \) |
| 19 | \( 1 + 4.63T + 19T^{2} \) |
| 23 | \( 1 + 7.11T + 23T^{2} \) |
| 29 | \( 1 + 0.128T + 29T^{2} \) |
| 37 | \( 1 - 8.42T + 37T^{2} \) |
| 41 | \( 1 + 7.37T + 41T^{2} \) |
| 43 | \( 1 - 0.230T + 43T^{2} \) |
| 47 | \( 1 + 8.03T + 47T^{2} \) |
| 53 | \( 1 - 5.73T + 53T^{2} \) |
| 59 | \( 1 - 9.50T + 59T^{2} \) |
| 61 | \( 1 + 7.84T + 61T^{2} \) |
| 67 | \( 1 - 4.82T + 67T^{2} \) |
| 71 | \( 1 + 3.40T + 71T^{2} \) |
| 73 | \( 1 - 2.69T + 73T^{2} \) |
| 79 | \( 1 - 4.52T + 79T^{2} \) |
| 83 | \( 1 - 2.67T + 83T^{2} \) |
| 89 | \( 1 + 2.20T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.08831747820017889988641052594, −8.527690923845406681771740102543, −7.53886823509137172058881431318, −6.57312195875029787734821867371, −6.06306356555725304644224355568, −4.95524972078817157893484169085, −4.49168103786802189546101103512, −3.75412114503109892793723639524, −2.15009586201185655449189305626, 0,
2.15009586201185655449189305626, 3.75412114503109892793723639524, 4.49168103786802189546101103512, 4.95524972078817157893484169085, 6.06306356555725304644224355568, 6.57312195875029787734821867371, 7.53886823509137172058881431318, 8.527690923845406681771740102543, 10.08831747820017889988641052594
