| L(s) = 1 | + 0.496·2-s − 1.75·4-s − 1.64·5-s − 3.34·7-s − 1.86·8-s − 0.815·10-s − 0.269·11-s − 6.61·13-s − 1.66·14-s + 2.58·16-s − 2.98·17-s + 5.96·19-s + 2.87·20-s − 0.133·22-s − 8.36·23-s − 2.30·25-s − 3.28·26-s + 5.86·28-s + 9.38·29-s − 0.868·31-s + 5.00·32-s − 1.48·34-s + 5.48·35-s + 11.6·37-s + 2.96·38-s + 3.05·40-s − 2.76·41-s + ⋯ |
| L(s) = 1 | + 0.351·2-s − 0.876·4-s − 0.734·5-s − 1.26·7-s − 0.658·8-s − 0.257·10-s − 0.0811·11-s − 1.83·13-s − 0.443·14-s + 0.645·16-s − 0.725·17-s + 1.36·19-s + 0.643·20-s − 0.0284·22-s − 1.74·23-s − 0.461·25-s − 0.643·26-s + 1.10·28-s + 1.74·29-s − 0.156·31-s + 0.885·32-s − 0.254·34-s + 0.927·35-s + 1.91·37-s + 0.480·38-s + 0.483·40-s − 0.432·41-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
| \(L(1)\) |
\(\approx\) |
\(0.5462656604\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5462656604\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.496T + 2T^{2} \) |
| 5 | \( 1 + 1.64T + 5T^{2} \) |
| 7 | \( 1 + 3.34T + 7T^{2} \) |
| 11 | \( 1 + 0.269T + 11T^{2} \) |
| 13 | \( 1 + 6.61T + 13T^{2} \) |
| 17 | \( 1 + 2.98T + 17T^{2} \) |
| 19 | \( 1 - 5.96T + 19T^{2} \) |
| 23 | \( 1 + 8.36T + 23T^{2} \) |
| 29 | \( 1 - 9.38T + 29T^{2} \) |
| 31 | \( 1 + 0.868T + 31T^{2} \) |
| 37 | \( 1 - 11.6T + 37T^{2} \) |
| 41 | \( 1 + 2.76T + 41T^{2} \) |
| 43 | \( 1 - 7.87T + 43T^{2} \) |
| 47 | \( 1 + 7.11T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 + 4.39T + 59T^{2} \) |
| 61 | \( 1 - 7.35T + 61T^{2} \) |
| 67 | \( 1 - 4.25T + 67T^{2} \) |
| 71 | \( 1 + 2.21T + 71T^{2} \) |
| 73 | \( 1 - 5.74T + 73T^{2} \) |
| 79 | \( 1 + 3.09T + 79T^{2} \) |
| 83 | \( 1 - 0.729T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 + 7.37T + 97T^{2} \) |
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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
−9.416336540311895777478772665208, −8.123788698210146384471768574157, −7.69111635570456518169396655787, −6.65655779507837123043003158638, −5.89813367684145030719058083357, −4.86864159363525885888430980260, −4.26729261542868704100003197608, −3.35187663871206362049987156237, −2.55618893961075310798613150260, −0.43518572569401739192468730358,
0.43518572569401739192468730358, 2.55618893961075310798613150260, 3.35187663871206362049987156237, 4.26729261542868704100003197608, 4.86864159363525885888430980260, 5.89813367684145030719058083357, 6.65655779507837123043003158638, 7.69111635570456518169396655787, 8.123788698210146384471768574157, 9.416336540311895777478772665208
