L(s) = 1 | − 1.44i·3-s + 0.898·9-s + 0.550i·11-s − 7.89·17-s − 8.34i·19-s − 5.65i·27-s + 0.797·33-s + 12.7·41-s − 10i·43-s − 7·49-s + 11.4i·51-s − 12.1·57-s − 6i·59-s − 14.3i·67-s − 13.6·73-s + ⋯ |
L(s) = 1 | − 0.836i·3-s + 0.299·9-s + 0.165i·11-s − 1.91·17-s − 1.91i·19-s − 1.08i·27-s + 0.138·33-s + 1.99·41-s − 1.52i·43-s − 49-s + 1.60i·51-s − 1.60·57-s − 0.781i·59-s − 1.75i·67-s − 1.60·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.236371791\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.236371791\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 1.44iT - 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 0.550iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 7.89T + 17T^{2} \) |
| 19 | \( 1 + 8.34iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 12.7T + 41T^{2} \) |
| 43 | \( 1 + 10iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 14.3iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 11.4iT - 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−9.052170775609728684513233151156, −8.317289485323344187913193582376, −7.24312141000043432129456574263, −6.89635390597825684759990802923, −6.09127159967759435054672510756, −4.84258022410827206373677875929, −4.19537219932644720003695052693, −2.71185581064950569311684051030, −1.91519532285394232707928963630, −0.46999951498999245449660171786,
1.56787672600461043390312226791, 2.86483721348894094557734912271, 4.08664443615262597745045310805, 4.42653700757017734927518446000, 5.61092136861478168337222489920, 6.37979273506460949545321030173, 7.34967322594072463007182652433, 8.228509010537515459924623850771, 9.049499471334203509275259918242, 9.712777573139476191852720978586