L(s) = 1 | − 2-s + 2.39·3-s + 4-s + 1.79·5-s − 2.39·6-s − 7-s − 8-s + 2.74·9-s − 1.79·10-s + 4.02·11-s + 2.39·12-s − 4.04·13-s + 14-s + 4.29·15-s + 16-s − 2.85·17-s − 2.74·18-s + 7.25·19-s + 1.79·20-s − 2.39·21-s − 4.02·22-s − 3.34·23-s − 2.39·24-s − 1.78·25-s + 4.04·26-s − 0.610·27-s − 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.38·3-s + 0.5·4-s + 0.801·5-s − 0.978·6-s − 0.377·7-s − 0.353·8-s + 0.915·9-s − 0.566·10-s + 1.21·11-s + 0.691·12-s − 1.12·13-s + 0.267·14-s + 1.10·15-s + 0.250·16-s − 0.691·17-s − 0.647·18-s + 1.66·19-s + 0.400·20-s − 0.523·21-s − 0.858·22-s − 0.696·23-s − 0.489·24-s − 0.357·25-s + 0.792·26-s − 0.117·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.896743008\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.896743008\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 - T \) |
good | 3 | \( 1 - 2.39T + 3T^{2} \) |
| 5 | \( 1 - 1.79T + 5T^{2} \) |
| 11 | \( 1 - 4.02T + 11T^{2} \) |
| 13 | \( 1 + 4.04T + 13T^{2} \) |
| 17 | \( 1 + 2.85T + 17T^{2} \) |
| 19 | \( 1 - 7.25T + 19T^{2} \) |
| 23 | \( 1 + 3.34T + 23T^{2} \) |
| 29 | \( 1 - 9.46T + 29T^{2} \) |
| 31 | \( 1 - 5.02T + 31T^{2} \) |
| 37 | \( 1 - 2.96T + 37T^{2} \) |
| 41 | \( 1 + 0.0546T + 41T^{2} \) |
| 43 | \( 1 - 2.62T + 43T^{2} \) |
| 47 | \( 1 + 5.09T + 47T^{2} \) |
| 53 | \( 1 - 6.76T + 53T^{2} \) |
| 59 | \( 1 - 2.56T + 59T^{2} \) |
| 61 | \( 1 + 5.88T + 61T^{2} \) |
| 67 | \( 1 + 7.03T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 - 16.6T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 - 3.21T + 83T^{2} \) |
| 89 | \( 1 + 3.96T + 89T^{2} \) |
| 97 | \( 1 - 1.99T + 97T^{2} \) |
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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
−8.092677629595381440005658244711, −7.62546301602840147235309847031, −6.72974226871416845954210574512, −6.29750986974879993511201132921, −5.23524565699603642589104544205, −4.24365721879309426492179818891, −3.33651468774425929570545223533, −2.60269640555005906100643718085, −2.00828766628705601697018006516, −0.951335473960372593942065499980,
0.951335473960372593942065499980, 2.00828766628705601697018006516, 2.60269640555005906100643718085, 3.33651468774425929570545223533, 4.24365721879309426492179818891, 5.23524565699603642589104544205, 6.29750986974879993511201132921, 6.72974226871416845954210574512, 7.62546301602840147235309847031, 8.092677629595381440005658244711