| L(s) = 1 | + 3-s + 2·5-s − 2·9-s − 11-s + 13-s + 2·15-s − 4·17-s + 4·19-s + 5·23-s − 25-s − 5·27-s + 6·29-s + 11·31-s − 33-s + 7·37-s + 39-s − 3·41-s − 2·43-s − 4·45-s + 3·47-s − 4·51-s − 2·53-s − 2·55-s + 4·57-s + 4·59-s − 5·61-s + 2·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s − 2/3·9-s − 0.301·11-s + 0.277·13-s + 0.516·15-s − 0.970·17-s + 0.917·19-s + 1.04·23-s − 1/5·25-s − 0.962·27-s + 1.11·29-s + 1.97·31-s − 0.174·33-s + 1.15·37-s + 0.160·39-s − 0.468·41-s − 0.304·43-s − 0.596·45-s + 0.437·47-s − 0.560·51-s − 0.274·53-s − 0.269·55-s + 0.529·57-s + 0.520·59-s − 0.640·61-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.006371642\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.006371642\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 11 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−16.70545761774839, −15.91329485594505, −15.43717045039437, −14.90446702331075, −14.12345659745076, −13.72786927305081, −13.41967152929456, −12.76418353686285, −11.87183509250482, −11.41067687993436, −10.74712175699657, −9.955736176687172, −9.614450871181980, −8.803955395470783, −8.452273642198880, −7.739183696746577, −6.903674674496420, −6.246399769490963, −5.683921999506796, −4.906334518214610, −4.229380972892053, −2.985031669713849, −2.802909996075694, −1.841175200430243, −0.7949068731545678,
0.7949068731545678, 1.841175200430243, 2.802909996075694, 2.985031669713849, 4.229380972892053, 4.906334518214610, 5.683921999506796, 6.246399769490963, 6.903674674496420, 7.739183696746577, 8.452273642198880, 8.803955395470783, 9.614450871181980, 9.955736176687172, 10.74712175699657, 11.41067687993436, 11.87183509250482, 12.76418353686285, 13.41967152929456, 13.72786927305081, 14.12345659745076, 14.90446702331075, 15.43717045039437, 15.91329485594505, 16.70545761774839
