| L(s) = 1 | − 3.16·3-s + 5-s + 3.16·7-s + 7.00·9-s + 6·13-s − 3.16·15-s − 2·17-s − 6.32·19-s − 10.0·21-s + 3.16·23-s + 25-s − 12.6·27-s − 4·29-s + 6.32·31-s + 3.16·35-s + 2·37-s − 18.9·39-s + 3.16·43-s + 7.00·45-s − 9.48·47-s + 3.00·49-s + 6.32·51-s + 6·53-s + 20.0·57-s + 6.32·59-s + 2·61-s + 22.1·63-s + ⋯ |
| L(s) = 1 | − 1.82·3-s + 0.447·5-s + 1.19·7-s + 2.33·9-s + 1.66·13-s − 0.816·15-s − 0.485·17-s − 1.45·19-s − 2.18·21-s + 0.659·23-s + 0.200·25-s − 2.43·27-s − 0.742·29-s + 1.13·31-s + 0.534·35-s + 0.328·37-s − 3.03·39-s + 0.482·43-s + 1.04·45-s − 1.38·47-s + 0.428·49-s + 0.885·51-s + 0.824·53-s + 2.64·57-s + 0.823·59-s + 0.256·61-s + 2.78·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.185742339\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.185742339\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 3 | \( 1 + 3.16T + 3T^{2} \) |
| 7 | \( 1 - 3.16T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 6.32T + 19T^{2} \) |
| 23 | \( 1 - 3.16T + 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 - 6.32T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 3.16T + 43T^{2} \) |
| 47 | \( 1 + 9.48T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 6.32T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 9.48T + 67T^{2} \) |
| 71 | \( 1 - 6.32T + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 - 3.16T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.993989183966640743854798695329, −8.806452491469126879548474285117, −8.063211486843389197476564753252, −6.80508645682960614151316738021, −6.29472436855216926146874384811, −5.51294510289470159195864801378, −4.75069261522851296956854782525, −3.98301113428580727168680938402, −1.94604474015196483810973752484, −0.939645492925122803991613819594,
0.939645492925122803991613819594, 1.94604474015196483810973752484, 3.98301113428580727168680938402, 4.75069261522851296956854782525, 5.51294510289470159195864801378, 6.29472436855216926146874384811, 6.80508645682960614151316738021, 8.063211486843389197476564753252, 8.806452491469126879548474285117, 9.993989183966640743854798695329
