| L(s) = 1 | − 4.24·5-s − 3·9-s − 1.41·13-s + 8·17-s + 12.9·25-s + 9.89·29-s − 7.07·37-s + 8·41-s + 12.7·45-s − 7·49-s + 7.07·53-s + 1.41·61-s + 6·65-s + 6·73-s + 9·81-s − 33.9·85-s − 10·89-s + 8·97-s + 15.5·101-s − 9.89·109-s + 14·113-s + 4.24·117-s + ⋯ |
| L(s) = 1 | − 1.89·5-s − 9-s − 0.392·13-s + 1.94·17-s + 2.59·25-s + 1.83·29-s − 1.16·37-s + 1.24·41-s + 1.89·45-s − 49-s + 0.971·53-s + 0.181·61-s + 0.744·65-s + 0.702·73-s + 81-s − 3.68·85-s − 1.05·89-s + 0.812·97-s + 1.54·101-s − 0.948·109-s + 1.31·113-s + 0.392·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9201383490\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9201383490\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 + 4.24T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 - 8T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 9.89T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 7.07T + 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 7.07T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 1.41T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−10.06259591774676199002124227601, −8.865125622339906784576385647604, −8.143928346229750537498379152982, −7.66833587467177388970112178367, −6.74279602406172279582898446152, −5.52424595302336136628923988633, −4.62536858611757736826292644150, −3.55783073996109915973187528046, −2.90254016079512524749074998128, −0.73296633131290248674811903317,
0.73296633131290248674811903317, 2.90254016079512524749074998128, 3.55783073996109915973187528046, 4.62536858611757736826292644150, 5.52424595302336136628923988633, 6.74279602406172279582898446152, 7.66833587467177388970112178367, 8.143928346229750537498379152982, 8.865125622339906784576385647604, 10.06259591774676199002124227601
