L(s) = 1 | + 2.90·3-s + 3.90·5-s + 5.44·9-s + 2.70·11-s − 1.75·13-s + 11.3·15-s + 17-s − 3.74·19-s − 8.40·23-s + 10.2·25-s + 7.10·27-s − 8.96·29-s + 7.25·31-s + 7.86·33-s + 6.20·37-s − 5.10·39-s + 10.7·41-s + 0.849·43-s + 21.2·45-s + 0.661·47-s + 2.90·51-s − 3.29·53-s + 10.5·55-s − 10.8·57-s − 3.24·59-s + 2.39·61-s − 6.85·65-s + ⋯ |
L(s) = 1 | + 1.67·3-s + 1.74·5-s + 1.81·9-s + 0.815·11-s − 0.486·13-s + 2.93·15-s + 0.242·17-s − 0.858·19-s − 1.75·23-s + 2.05·25-s + 1.36·27-s − 1.66·29-s + 1.30·31-s + 1.36·33-s + 1.01·37-s − 0.816·39-s + 1.68·41-s + 0.129·43-s + 3.17·45-s + 0.0964·47-s + 0.406·51-s − 0.452·53-s + 1.42·55-s − 1.43·57-s − 0.422·59-s + 0.306·61-s − 0.850·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.940188787\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.940188787\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 3 | \( 1 - 2.90T + 3T^{2} \) |
| 5 | \( 1 - 3.90T + 5T^{2} \) |
| 11 | \( 1 - 2.70T + 11T^{2} \) |
| 13 | \( 1 + 1.75T + 13T^{2} \) |
| 19 | \( 1 + 3.74T + 19T^{2} \) |
| 23 | \( 1 + 8.40T + 23T^{2} \) |
| 29 | \( 1 + 8.96T + 29T^{2} \) |
| 31 | \( 1 - 7.25T + 31T^{2} \) |
| 37 | \( 1 - 6.20T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 - 0.849T + 43T^{2} \) |
| 47 | \( 1 - 0.661T + 47T^{2} \) |
| 53 | \( 1 + 3.29T + 53T^{2} \) |
| 59 | \( 1 + 3.24T + 59T^{2} \) |
| 61 | \( 1 - 2.39T + 61T^{2} \) |
| 67 | \( 1 - 0.395T + 67T^{2} \) |
| 71 | \( 1 + 5.99T + 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 + 6.84T + 79T^{2} \) |
| 83 | \( 1 - 0.0591T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 + 8.01T + 97T^{2} \) |
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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
−8.854932841959153299252397830052, −7.973881011495045699437216948432, −7.31875874322532259959554552946, −6.25412148272960806623172858556, −5.87597395221082465539379808694, −4.55778194744361890050502681971, −3.86086539055704164385073523491, −2.72162181998729366223585578488, −2.18189746427051444577502864359, −1.45410253477311405757863759600,
1.45410253477311405757863759600, 2.18189746427051444577502864359, 2.72162181998729366223585578488, 3.86086539055704164385073523491, 4.55778194744361890050502681971, 5.87597395221082465539379808694, 6.25412148272960806623172858556, 7.31875874322532259959554552946, 7.973881011495045699437216948432, 8.854932841959153299252397830052