L(s) = 1 | + 0.141·2-s − 1.97·4-s − 0.858·7-s − 0.563·8-s + 3.67·11-s + 4.58·13-s − 0.121·14-s + 3.87·16-s + 5.30·17-s + 6.36·19-s + 0.521·22-s − 3.42·23-s + 0.649·26-s + 1.69·28-s − 3.73·29-s + 1.25·31-s + 1.67·32-s + 0.751·34-s − 7.45·37-s + 0.902·38-s + 2.53·41-s − 3.37·43-s − 7.28·44-s − 0.485·46-s + 8.49·47-s − 6.26·49-s − 9.07·52-s + ⋯ |
L(s) = 1 | + 0.100·2-s − 0.989·4-s − 0.324·7-s − 0.199·8-s + 1.10·11-s + 1.27·13-s − 0.0325·14-s + 0.969·16-s + 1.28·17-s + 1.46·19-s + 0.111·22-s − 0.713·23-s + 0.127·26-s + 0.321·28-s − 0.693·29-s + 0.225·31-s + 0.296·32-s + 0.128·34-s − 1.22·37-s + 0.146·38-s + 0.395·41-s − 0.515·43-s − 1.09·44-s − 0.0715·46-s + 1.23·47-s − 0.894·49-s − 1.25·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.899262950\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.899262950\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 0.141T + 2T^{2} \) |
| 7 | \( 1 + 0.858T + 7T^{2} \) |
| 11 | \( 1 - 3.67T + 11T^{2} \) |
| 13 | \( 1 - 4.58T + 13T^{2} \) |
| 17 | \( 1 - 5.30T + 17T^{2} \) |
| 19 | \( 1 - 6.36T + 19T^{2} \) |
| 23 | \( 1 + 3.42T + 23T^{2} \) |
| 29 | \( 1 + 3.73T + 29T^{2} \) |
| 31 | \( 1 - 1.25T + 31T^{2} \) |
| 37 | \( 1 + 7.45T + 37T^{2} \) |
| 41 | \( 1 - 2.53T + 41T^{2} \) |
| 43 | \( 1 + 3.37T + 43T^{2} \) |
| 47 | \( 1 - 8.49T + 47T^{2} \) |
| 53 | \( 1 - 2.34T + 53T^{2} \) |
| 59 | \( 1 + 13.1T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 - 3.34T + 67T^{2} \) |
| 71 | \( 1 - 4.32T + 71T^{2} \) |
| 73 | \( 1 + 9.08T + 73T^{2} \) |
| 79 | \( 1 + 3.24T + 79T^{2} \) |
| 83 | \( 1 - 7.39T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−8.184941616684401270016952832355, −7.54749997491785807141597031401, −6.62963968388117209625920535925, −5.81020679726058978732641431457, −5.40147698523802868315604294169, −4.35517004591866126745666461012, −3.56718245557094863501175449574, −3.29261104145051872889558258509, −1.58269953710054108240292935980, −0.801452305009849189091906986947,
0.801452305009849189091906986947, 1.58269953710054108240292935980, 3.29261104145051872889558258509, 3.56718245557094863501175449574, 4.35517004591866126745666461012, 5.40147698523802868315604294169, 5.81020679726058978732641431457, 6.62963968388117209625920535925, 7.54749997491785807141597031401, 8.184941616684401270016952832355