L(s) = 1 | − 2·3-s − 2·4-s + 4·7-s + 9-s + 3·11-s + 4·12-s + 2·13-s + 4·16-s − 6·17-s − 8·21-s + 4·27-s − 8·28-s + 3·29-s + 7·31-s − 6·33-s − 2·36-s + 8·37-s − 4·39-s + 6·41-s + 4·43-s − 6·44-s − 6·47-s − 8·48-s + 9·49-s + 12·51-s − 4·52-s − 6·53-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 4-s + 1.51·7-s + 1/3·9-s + 0.904·11-s + 1.15·12-s + 0.554·13-s + 16-s − 1.45·17-s − 1.74·21-s + 0.769·27-s − 1.51·28-s + 0.557·29-s + 1.25·31-s − 1.04·33-s − 1/3·36-s + 1.31·37-s − 0.640·39-s + 0.937·41-s + 0.609·43-s − 0.904·44-s − 0.875·47-s − 1.15·48-s + 9/7·49-s + 1.68·51-s − 0.554·52-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.345912066\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.345912066\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−7.898934481127464840761347529461, −6.88228420629183365506661100954, −6.21420436019641569413437532011, −5.66365637265296072618464140836, −4.71701950926652242293967162188, −4.59471935191516245738409702732, −3.85962538046679493183061228257, −2.52209375977356831225573923389, −1.33939202709329241165983937159, −0.69341724549295096836045016557,
0.69341724549295096836045016557, 1.33939202709329241165983937159, 2.52209375977356831225573923389, 3.85962538046679493183061228257, 4.59471935191516245738409702732, 4.71701950926652242293967162188, 5.66365637265296072618464140836, 6.21420436019641569413437532011, 6.88228420629183365506661100954, 7.898934481127464840761347529461