| L(s) = 1 | + 1.56·3-s + 3.56·5-s − 1.56·7-s − 0.561·9-s − 3.12·11-s + 13-s + 5.56·15-s − 6.68·17-s + 3.12·19-s − 2.43·21-s + 8·23-s + 7.68·25-s − 5.56·27-s − 2·29-s − 4·31-s − 4.87·33-s − 5.56·35-s − 2.68·37-s + 1.56·39-s + 5.12·41-s − 9.56·43-s − 2·45-s + 12.6·47-s − 4.56·49-s − 10.4·51-s − 5.12·53-s − 11.1·55-s + ⋯ |
| L(s) = 1 | + 0.901·3-s + 1.59·5-s − 0.590·7-s − 0.187·9-s − 0.941·11-s + 0.277·13-s + 1.43·15-s − 1.62·17-s + 0.716·19-s − 0.532·21-s + 1.66·23-s + 1.53·25-s − 1.07·27-s − 0.371·29-s − 0.718·31-s − 0.848·33-s − 0.940·35-s − 0.441·37-s + 0.250·39-s + 0.800·41-s − 1.45·43-s − 0.298·45-s + 1.85·47-s − 0.651·49-s − 1.46·51-s − 0.703·53-s − 1.49·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.709639975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.709639975\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 1.56T + 3T^{2} \) |
| 5 | \( 1 - 3.56T + 5T^{2} \) |
| 7 | \( 1 + 1.56T + 7T^{2} \) |
| 11 | \( 1 + 3.12T + 11T^{2} \) |
| 17 | \( 1 + 6.68T + 17T^{2} \) |
| 19 | \( 1 - 3.12T + 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 2.68T + 37T^{2} \) |
| 41 | \( 1 - 5.12T + 41T^{2} \) |
| 43 | \( 1 + 9.56T + 43T^{2} \) |
| 47 | \( 1 - 12.6T + 47T^{2} \) |
| 53 | \( 1 + 5.12T + 53T^{2} \) |
| 59 | \( 1 - 3.12T + 59T^{2} \) |
| 61 | \( 1 - 2.87T + 61T^{2} \) |
| 67 | \( 1 + 3.12T + 67T^{2} \) |
| 71 | \( 1 + 4.68T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 8.24T + 97T^{2} \) |
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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
−13.04391640855858975115348081131, −11.20564381578193824547454095069, −10.26535946633354517547248924129, −9.224493573715851073183082124645, −8.817083197804876140368194104643, −7.29768299445890626896906427281, −6.14169966763019824908061051894, −5.10427054282190041927593936329, −3.14171025325337315454839249449, −2.16392535255847397846061755356,
2.16392535255847397846061755356, 3.14171025325337315454839249449, 5.10427054282190041927593936329, 6.14169966763019824908061051894, 7.29768299445890626896906427281, 8.817083197804876140368194104643, 9.224493573715851073183082124645, 10.26535946633354517547248924129, 11.20564381578193824547454095069, 13.04391640855858975115348081131
