| L(s) = 1 | + 0.711·2-s − 1.49·4-s − 0.309·5-s − 4.19·7-s − 2.48·8-s − 0.220·10-s − 1.06·11-s + 3.09·13-s − 2.98·14-s + 1.21·16-s − 1.63·17-s − 5.86·19-s + 0.462·20-s − 0.756·22-s + 6.91·23-s − 4.90·25-s + 2.20·26-s + 6.27·28-s + 7.66·29-s − 5.79·31-s + 5.83·32-s − 1.16·34-s + 1.30·35-s + 4.34·37-s − 4.17·38-s + 0.770·40-s + 11.0·41-s + ⋯ |
| L(s) = 1 | + 0.503·2-s − 0.746·4-s − 0.138·5-s − 1.58·7-s − 0.878·8-s − 0.0697·10-s − 0.320·11-s + 0.858·13-s − 0.798·14-s + 0.304·16-s − 0.396·17-s − 1.34·19-s + 0.103·20-s − 0.161·22-s + 1.44·23-s − 0.980·25-s + 0.432·26-s + 1.18·28-s + 1.42·29-s − 1.04·31-s + 1.03·32-s − 0.199·34-s + 0.219·35-s + 0.714·37-s − 0.677·38-s + 0.121·40-s + 1.72·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.072232706\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.072232706\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + T \) |
| good | 2 | \( 1 - 0.711T + 2T^{2} \) |
| 5 | \( 1 + 0.309T + 5T^{2} \) |
| 7 | \( 1 + 4.19T + 7T^{2} \) |
| 11 | \( 1 + 1.06T + 11T^{2} \) |
| 13 | \( 1 - 3.09T + 13T^{2} \) |
| 17 | \( 1 + 1.63T + 17T^{2} \) |
| 19 | \( 1 + 5.86T + 19T^{2} \) |
| 23 | \( 1 - 6.91T + 23T^{2} \) |
| 29 | \( 1 - 7.66T + 29T^{2} \) |
| 31 | \( 1 + 5.79T + 31T^{2} \) |
| 37 | \( 1 - 4.34T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 + 1.61T + 43T^{2} \) |
| 47 | \( 1 - 4.79T + 47T^{2} \) |
| 53 | \( 1 + 8.64T + 53T^{2} \) |
| 59 | \( 1 + 4.12T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 - 7.05T + 71T^{2} \) |
| 73 | \( 1 - 1.39T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 - 7.68T + 83T^{2} \) |
| 89 | \( 1 - 9.42T + 89T^{2} \) |
| 97 | \( 1 + 2.84T + 97T^{2} \) |
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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
−9.084914251354961105129781629828, −8.522819515735988664258261422077, −7.48166535579195768145267470871, −6.30336771863687725395786510313, −6.16747035106196841677013949356, −4.98721090804346451717653766389, −4.09726146132481910318017019338, −3.43636651578337442832335297312, −2.54958448916948856243325275073, −0.61003906585490433107842665113,
0.61003906585490433107842665113, 2.54958448916948856243325275073, 3.43636651578337442832335297312, 4.09726146132481910318017019338, 4.98721090804346451717653766389, 6.16747035106196841677013949356, 6.30336771863687725395786510313, 7.48166535579195768145267470871, 8.522819515735988664258261422077, 9.084914251354961105129781629828
