| L(s) = 1 | − 2.34·2-s + 3.51·4-s − 1.00·5-s − 3.03·7-s − 3.56·8-s + 2.35·10-s − 5.41·11-s + 5.93·13-s + 7.12·14-s + 1.34·16-s + 1.60·17-s + 4.16·19-s − 3.53·20-s + 12.7·22-s − 8.35·23-s − 3.99·25-s − 13.9·26-s − 10.6·28-s + 4.52·29-s − 6.03·31-s + 3.97·32-s − 3.76·34-s + 3.04·35-s − 6.51·37-s − 9.78·38-s + 3.58·40-s + 3.85·41-s + ⋯ |
| L(s) = 1 | − 1.66·2-s + 1.75·4-s − 0.448·5-s − 1.14·7-s − 1.26·8-s + 0.745·10-s − 1.63·11-s + 1.64·13-s + 1.90·14-s + 0.337·16-s + 0.388·17-s + 0.955·19-s − 0.790·20-s + 2.71·22-s − 1.74·23-s − 0.798·25-s − 2.73·26-s − 2.01·28-s + 0.840·29-s − 1.08·31-s + 0.702·32-s − 0.646·34-s + 0.514·35-s − 1.07·37-s − 1.58·38-s + 0.566·40-s + 0.601·41-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
| \(L(1)\) |
\(\approx\) |
\(0.3605181110\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3605181110\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.34T + 2T^{2} \) |
| 5 | \( 1 + 1.00T + 5T^{2} \) |
| 7 | \( 1 + 3.03T + 7T^{2} \) |
| 11 | \( 1 + 5.41T + 11T^{2} \) |
| 13 | \( 1 - 5.93T + 13T^{2} \) |
| 17 | \( 1 - 1.60T + 17T^{2} \) |
| 19 | \( 1 - 4.16T + 19T^{2} \) |
| 23 | \( 1 + 8.35T + 23T^{2} \) |
| 29 | \( 1 - 4.52T + 29T^{2} \) |
| 31 | \( 1 + 6.03T + 31T^{2} \) |
| 37 | \( 1 + 6.51T + 37T^{2} \) |
| 41 | \( 1 - 3.85T + 41T^{2} \) |
| 43 | \( 1 + 4.15T + 43T^{2} \) |
| 47 | \( 1 + 4.57T + 47T^{2} \) |
| 53 | \( 1 - 3.88T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 8.26T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 - 9.76T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 3.07T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 + 7.06T + 97T^{2} \) |
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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
−9.093230930296876806972152848727, −8.138132037755738022888865801748, −7.956942551360859515437652341019, −7.01921795534297040246200273625, −6.19488676387101299179659680109, −5.42665277095904370182076652568, −3.83524598541146929943364387899, −3.05678766813484007980575880806, −1.87758229163159449719252200415, −0.48803269594860323437092360751,
0.48803269594860323437092360751, 1.87758229163159449719252200415, 3.05678766813484007980575880806, 3.83524598541146929943364387899, 5.42665277095904370182076652568, 6.19488676387101299179659680109, 7.01921795534297040246200273625, 7.956942551360859515437652341019, 8.138132037755738022888865801748, 9.093230930296876806972152848727
