L(s) = 1 | + 4·7-s − 2·13-s + 8·19-s − 4·31-s + 10·37-s − 8·43-s + 9·49-s + 14·61-s + 16·67-s + 10·73-s − 4·79-s − 8·91-s − 14·97-s − 20·103-s + 2·109-s + ⋯ |
L(s) = 1 | + 1.51·7-s − 0.554·13-s + 1.83·19-s − 0.718·31-s + 1.64·37-s − 1.21·43-s + 9/7·49-s + 1.79·61-s + 1.95·67-s + 1.17·73-s − 0.450·79-s − 0.838·91-s − 1.42·97-s − 1.97·103-s + 0.191·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.881224702\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.881224702\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−10.02595561869790412734353017930, −9.351967908821189871222500143687, −8.239034030776179718908314632866, −7.71900234701090149632840343207, −6.84353614161574326267996849514, −5.45300314870917608103230012398, −4.98062769154464971467686882983, −3.83217595589008344467032618819, −2.47608687475654661435538320080, −1.22304458410378265801798735343,
1.22304458410378265801798735343, 2.47608687475654661435538320080, 3.83217595589008344467032618819, 4.98062769154464971467686882983, 5.45300314870917608103230012398, 6.84353614161574326267996849514, 7.71900234701090149632840343207, 8.239034030776179718908314632866, 9.351967908821189871222500143687, 10.02595561869790412734353017930