| L(s) = 1 | − 3·3-s + 14.6·5-s + 9·9-s + 11.4·11-s + 20.8·13-s − 43.9·15-s − 18.8·17-s + 91.0·19-s − 28.9·23-s + 89.3·25-s − 27·27-s + 281.·29-s − 276.·31-s − 34.3·33-s + 250.·37-s − 62.4·39-s − 12.1·41-s − 65.2·43-s + 131.·45-s + 199.·47-s + 56.4·51-s − 64.1·53-s + 167.·55-s − 273.·57-s − 69.0·59-s + 656.·61-s + 304.·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.30·5-s + 0.333·9-s + 0.313·11-s + 0.443·13-s − 0.756·15-s − 0.268·17-s + 1.09·19-s − 0.262·23-s + 0.715·25-s − 0.192·27-s + 1.80·29-s − 1.59·31-s − 0.181·33-s + 1.11·37-s − 0.256·39-s − 0.0462·41-s − 0.231·43-s + 0.436·45-s + 0.620·47-s + 0.155·51-s − 0.166·53-s + 0.410·55-s − 0.634·57-s − 0.152·59-s + 1.37·61-s + 0.581·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.739938897\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.739938897\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 14.6T + 125T^{2} \) |
| 11 | \( 1 - 11.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 20.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 18.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 91.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 28.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 281.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 276.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 250.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 12.1T + 6.89e4T^{2} \) |
| 43 | \( 1 + 65.2T + 7.95e4T^{2} \) |
| 47 | \( 1 - 199.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 64.1T + 1.48e5T^{2} \) |
| 59 | \( 1 + 69.0T + 2.05e5T^{2} \) |
| 61 | \( 1 - 656.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 419.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.02e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 103.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 260.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 879.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.48e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 177.T + 9.12e5T^{2} \) |
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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
−8.834439404436465735621568066058, −7.80369787697610890467946316732, −6.86917658769974071454426939715, −6.21006689578783724856102304888, −5.59240629846379069101675569230, −4.87554662771421914713538876669, −3.81501351750869924358437545032, −2.67330448953823047186293382137, −1.67312514306597313840634644695, −0.800498491916692887339731902263,
0.800498491916692887339731902263, 1.67312514306597313840634644695, 2.67330448953823047186293382137, 3.81501351750869924358437545032, 4.87554662771421914713538876669, 5.59240629846379069101675569230, 6.21006689578783724856102304888, 6.86917658769974071454426939715, 7.80369787697610890467946316732, 8.834439404436465735621568066058
