L(s) = 1 | − 2·3-s + 9-s − 11-s − 2·13-s + 6·17-s − 4·19-s + 2·23-s + 4·27-s − 10·29-s + 8·31-s + 2·33-s + 8·37-s + 4·39-s − 2·41-s − 2·47-s − 7·49-s − 12·51-s + 8·57-s + 12·59-s − 10·61-s + 6·67-s − 4·69-s + 6·73-s − 12·79-s − 11·81-s − 16·83-s + 20·87-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 1/3·9-s − 0.301·11-s − 0.554·13-s + 1.45·17-s − 0.917·19-s + 0.417·23-s + 0.769·27-s − 1.85·29-s + 1.43·31-s + 0.348·33-s + 1.31·37-s + 0.640·39-s − 0.312·41-s − 0.291·47-s − 49-s − 1.68·51-s + 1.05·57-s + 1.56·59-s − 1.28·61-s + 0.733·67-s − 0.481·69-s + 0.702·73-s − 1.35·79-s − 1.22·81-s − 1.75·83-s + 2.14·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−7.904485605411936550870921183333, −7.22017537567996611933167852237, −6.36445053083546889629036058102, −5.78828686553519150683753904037, −5.13538484446126840174890144527, −4.45578662938336781261107879972, −3.39342886502123065607142851851, −2.41829749235878729015251645544, −1.14050065474498014904255046083, 0,
1.14050065474498014904255046083, 2.41829749235878729015251645544, 3.39342886502123065607142851851, 4.45578662938336781261107879972, 5.13538484446126840174890144527, 5.78828686553519150683753904037, 6.36445053083546889629036058102, 7.22017537567996611933167852237, 7.904485605411936550870921183333