| L(s) = 1 | − 2.73·3-s + 3.07·5-s + 4.47·9-s − 0.693·11-s − 1.77·13-s − 8.40·15-s − 1.86·17-s − 2.98·19-s − 23-s + 4.45·25-s − 4.04·27-s + 2.84·29-s + 1.48·31-s + 1.89·33-s + 1.47·37-s + 4.85·39-s + 10.3·41-s + 0.231·43-s + 13.7·45-s − 7.58·47-s + 5.10·51-s + 4.64·53-s − 2.13·55-s + 8.17·57-s − 8.95·59-s + 2.24·61-s − 5.45·65-s + ⋯ |
| L(s) = 1 | − 1.57·3-s + 1.37·5-s + 1.49·9-s − 0.209·11-s − 0.492·13-s − 2.17·15-s − 0.452·17-s − 0.685·19-s − 0.208·23-s + 0.890·25-s − 0.777·27-s + 0.527·29-s + 0.267·31-s + 0.329·33-s + 0.242·37-s + 0.777·39-s + 1.61·41-s + 0.0353·43-s + 2.05·45-s − 1.10·47-s + 0.714·51-s + 0.638·53-s − 0.287·55-s + 1.08·57-s − 1.16·59-s + 0.287·61-s − 0.676·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 2.73T + 3T^{2} \) |
| 5 | \( 1 - 3.07T + 5T^{2} \) |
| 11 | \( 1 + 0.693T + 11T^{2} \) |
| 13 | \( 1 + 1.77T + 13T^{2} \) |
| 17 | \( 1 + 1.86T + 17T^{2} \) |
| 19 | \( 1 + 2.98T + 19T^{2} \) |
| 29 | \( 1 - 2.84T + 29T^{2} \) |
| 31 | \( 1 - 1.48T + 31T^{2} \) |
| 37 | \( 1 - 1.47T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 - 0.231T + 43T^{2} \) |
| 47 | \( 1 + 7.58T + 47T^{2} \) |
| 53 | \( 1 - 4.64T + 53T^{2} \) |
| 59 | \( 1 + 8.95T + 59T^{2} \) |
| 61 | \( 1 - 2.24T + 61T^{2} \) |
| 67 | \( 1 - 2.85T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 + 3.50T + 73T^{2} \) |
| 79 | \( 1 - 7.42T + 79T^{2} \) |
| 83 | \( 1 + 5.90T + 83T^{2} \) |
| 89 | \( 1 + 17.1T + 89T^{2} \) |
| 97 | \( 1 + 4.64T + 97T^{2} \) |
| 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
−7.07159504642029251114800635559, −6.48283352640204236571313891105, −5.97510211027437046146971979146, −5.51442875733973534786770713676, −4.76432140753734612170555675320, −4.25281798795828479619200203143, −2.83101518048802346055240516513, −2.05027167062098694276930915253, −1.13754501721886596371629827641, 0,
1.13754501721886596371629827641, 2.05027167062098694276930915253, 2.83101518048802346055240516513, 4.25281798795828479619200203143, 4.76432140753734612170555675320, 5.51442875733973534786770713676, 5.97510211027437046146971979146, 6.48283352640204236571313891105, 7.07159504642029251114800635559
