| L(s) = 1 | − 1.51·2-s + 3-s + 0.291·4-s − 3.80·5-s − 1.51·6-s + 2.58·8-s + 9-s + 5.76·10-s + 1.15·11-s + 0.291·12-s + 4.56·13-s − 3.80·15-s − 4.49·16-s + 6.92·17-s − 1.51·18-s − 2.40·19-s − 1.11·20-s − 1.75·22-s + 3.28·23-s + 2.58·24-s + 9.47·25-s − 6.90·26-s + 27-s + 1.28·29-s + 5.76·30-s + 0.334·31-s + 1.63·32-s + ⋯ |
| L(s) = 1 | − 1.07·2-s + 0.577·3-s + 0.145·4-s − 1.70·5-s − 0.618·6-s + 0.914·8-s + 0.333·9-s + 1.82·10-s + 0.348·11-s + 0.0842·12-s + 1.26·13-s − 0.982·15-s − 1.12·16-s + 1.67·17-s − 0.356·18-s − 0.551·19-s − 0.248·20-s − 0.373·22-s + 0.684·23-s + 0.527·24-s + 1.89·25-s − 1.35·26-s + 0.192·27-s + 0.237·29-s + 1.05·30-s + 0.0601·31-s + 0.289·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.054810800\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.054810800\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 1.51T + 2T^{2} \) |
| 5 | \( 1 + 3.80T + 5T^{2} \) |
| 11 | \( 1 - 1.15T + 11T^{2} \) |
| 13 | \( 1 - 4.56T + 13T^{2} \) |
| 17 | \( 1 - 6.92T + 17T^{2} \) |
| 19 | \( 1 + 2.40T + 19T^{2} \) |
| 23 | \( 1 - 3.28T + 23T^{2} \) |
| 29 | \( 1 - 1.28T + 29T^{2} \) |
| 31 | \( 1 - 0.334T + 31T^{2} \) |
| 37 | \( 1 - 1.13T + 37T^{2} \) |
| 43 | \( 1 - 2.12T + 43T^{2} \) |
| 47 | \( 1 - 7.39T + 47T^{2} \) |
| 53 | \( 1 + 2.25T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 + 2.57T + 61T^{2} \) |
| 67 | \( 1 + 6.21T + 67T^{2} \) |
| 71 | \( 1 - 2.30T + 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 + 2.58T + 83T^{2} \) |
| 89 | \( 1 - 9.21T + 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
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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
−8.110018200936449556583158731680, −7.62550289804334851928003002546, −7.16597524928568455439486334737, −6.18163077848147230403467856515, −5.03602572938532498351463043348, −4.10768504461385237863519027302, −3.74432385229345803056199990238, −2.87126707878407130751757385625, −1.38766469162176339825603674524, −0.70645246722566756506017658707,
0.70645246722566756506017658707, 1.38766469162176339825603674524, 2.87126707878407130751757385625, 3.74432385229345803056199990238, 4.10768504461385237863519027302, 5.03602572938532498351463043348, 6.18163077848147230403467856515, 7.16597524928568455439486334737, 7.62550289804334851928003002546, 8.110018200936449556583158731680
