L(s) = 1 | − 3-s + 5-s − 3·7-s + 9-s − 4.87·11-s − 2.87·13-s − 15-s + 3.87·17-s + 2.87·19-s + 3·21-s + 23-s + 25-s − 27-s − 5.87·29-s − 3·31-s + 4.87·33-s − 3·35-s + 37-s + 2.87·39-s − 5.87·41-s − 3.74·43-s + 45-s + 6.87·47-s + 2·49-s − 3.87·51-s − 3.87·53-s − 4.87·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.13·7-s + 0.333·9-s − 1.46·11-s − 0.796·13-s − 0.258·15-s + 0.939·17-s + 0.659·19-s + 0.654·21-s + 0.208·23-s + 0.200·25-s − 0.192·27-s − 1.09·29-s − 0.538·31-s + 0.848·33-s − 0.507·35-s + 0.164·37-s + 0.460·39-s − 0.917·41-s − 0.571·43-s + 0.149·45-s + 1.00·47-s + 0.285·49-s − 0.542·51-s − 0.531·53-s − 0.657·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8658449383\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8658449383\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 + 4.87T + 11T^{2} \) |
| 13 | \( 1 + 2.87T + 13T^{2} \) |
| 17 | \( 1 - 3.87T + 17T^{2} \) |
| 19 | \( 1 - 2.87T + 19T^{2} \) |
| 29 | \( 1 + 5.87T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 + 5.87T + 41T^{2} \) |
| 43 | \( 1 + 3.74T + 43T^{2} \) |
| 47 | \( 1 - 6.87T + 47T^{2} \) |
| 53 | \( 1 + 3.87T + 53T^{2} \) |
| 59 | \( 1 + 5.87T + 59T^{2} \) |
| 61 | \( 1 - 8.87T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 + 2.87T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 0.127T + 83T^{2} \) |
| 89 | \( 1 - 1.74T + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−7.934648345611399691810393431830, −7.38894696416175457916750491779, −6.74937865689288564611238891180, −5.80261630818328094586560056672, −5.43790872186153109405999883788, −4.73802348953620290143359012394, −3.50002572276429566472762778413, −2.90257522491211421159968826203, −1.90547043926583619394262736289, −0.48990319060768075135299499052,
0.48990319060768075135299499052, 1.90547043926583619394262736289, 2.90257522491211421159968826203, 3.50002572276429566472762778413, 4.73802348953620290143359012394, 5.43790872186153109405999883788, 5.80261630818328094586560056672, 6.74937865689288564611238891180, 7.38894696416175457916750491779, 7.934648345611399691810393431830