L(s) = 1 | − 3-s + 5-s − 2·9-s + 6·11-s − 2·13-s − 15-s + 6·17-s − 8·19-s + 3·23-s + 25-s + 5·27-s + 3·29-s − 2·31-s − 6·33-s + 8·37-s + 2·39-s + 3·41-s + 5·43-s − 2·45-s − 6·51-s + 12·53-s + 6·55-s + 8·57-s + 61-s − 2·65-s − 7·67-s − 3·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 2/3·9-s + 1.80·11-s − 0.554·13-s − 0.258·15-s + 1.45·17-s − 1.83·19-s + 0.625·23-s + 1/5·25-s + 0.962·27-s + 0.557·29-s − 0.359·31-s − 1.04·33-s + 1.31·37-s + 0.320·39-s + 0.468·41-s + 0.762·43-s − 0.298·45-s − 0.840·51-s + 1.64·53-s + 0.809·55-s + 1.05·57-s + 0.128·61-s − 0.248·65-s − 0.855·67-s − 0.361·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.445252028\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.445252028\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−10.02709499826263992686786819872, −9.160354239682438919186405398338, −8.526752423181032766214386517330, −7.33381264956548829537687263175, −6.35483959490873446885572231998, −5.90403452102478988877219415616, −4.79699753647159991331019839993, −3.79089948904898683611492742798, −2.47880780691297269804291393924, −1.01545008916300809750400011716,
1.01545008916300809750400011716, 2.47880780691297269804291393924, 3.79089948904898683611492742798, 4.79699753647159991331019839993, 5.90403452102478988877219415616, 6.35483959490873446885572231998, 7.33381264956548829537687263175, 8.526752423181032766214386517330, 9.160354239682438919186405398338, 10.02709499826263992686786819872