L(s) = 1 | + 0.822·2-s − 0.955·3-s − 1.32·4-s − 3.05·5-s − 0.785·6-s − 2.73·8-s − 2.08·9-s − 2.51·10-s − 6.35·11-s + 1.26·12-s + 3.93·13-s + 2.91·15-s + 0.402·16-s − 2.97·17-s − 1.71·18-s − 5.53·19-s + 4.04·20-s − 5.22·22-s − 5.46·23-s + 2.60·24-s + 4.34·25-s + 3.23·26-s + 4.85·27-s − 5.86·29-s + 2.39·30-s + 10.6·31-s + 5.79·32-s + ⋯ |
L(s) = 1 | + 0.581·2-s − 0.551·3-s − 0.662·4-s − 1.36·5-s − 0.320·6-s − 0.966·8-s − 0.695·9-s − 0.794·10-s − 1.91·11-s + 0.365·12-s + 1.09·13-s + 0.753·15-s + 0.100·16-s − 0.720·17-s − 0.404·18-s − 1.27·19-s + 0.905·20-s − 1.11·22-s − 1.13·23-s + 0.532·24-s + 0.868·25-s + 0.633·26-s + 0.935·27-s − 1.08·29-s + 0.438·30-s + 1.91·31-s + 1.02·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2491397283\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2491397283\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 2 | \( 1 - 0.822T + 2T^{2} \) |
| 3 | \( 1 + 0.955T + 3T^{2} \) |
| 5 | \( 1 + 3.05T + 5T^{2} \) |
| 11 | \( 1 + 6.35T + 11T^{2} \) |
| 13 | \( 1 - 3.93T + 13T^{2} \) |
| 17 | \( 1 + 2.97T + 17T^{2} \) |
| 19 | \( 1 + 5.53T + 19T^{2} \) |
| 23 | \( 1 + 5.46T + 23T^{2} \) |
| 29 | \( 1 + 5.86T + 29T^{2} \) |
| 31 | \( 1 - 10.6T + 31T^{2} \) |
| 37 | \( 1 + 0.638T + 37T^{2} \) |
| 43 | \( 1 + 0.978T + 43T^{2} \) |
| 47 | \( 1 + 3.60T + 47T^{2} \) |
| 53 | \( 1 + 2.10T + 53T^{2} \) |
| 59 | \( 1 + 5.71T + 59T^{2} \) |
| 61 | \( 1 - 6.61T + 61T^{2} \) |
| 67 | \( 1 - 2.82T + 67T^{2} \) |
| 71 | \( 1 + 0.220T + 71T^{2} \) |
| 73 | \( 1 + 6.59T + 73T^{2} \) |
| 79 | \( 1 - 1.73T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 - 2.25T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−8.858666677174237628494360091681, −8.196499017023542813599154139685, −7.943013529500692506857709213264, −6.52226815206138186153360471249, −5.86164472000291288233353722503, −4.99944730870728909330690883510, −4.32460599547909337663200137126, −3.53840792253679266612180168874, −2.55673224513536652692280122245, −0.29301267786317257652006075403,
0.29301267786317257652006075403, 2.55673224513536652692280122245, 3.53840792253679266612180168874, 4.32460599547909337663200137126, 4.99944730870728909330690883510, 5.86164472000291288233353722503, 6.52226815206138186153360471249, 7.943013529500692506857709213264, 8.196499017023542813599154139685, 8.858666677174237628494360091681