| L(s) = 1 | − 2.43·2-s + 3.91·4-s − 4.64·8-s − 6.50·11-s + 3.47·16-s + 15.8·22-s − 6.62·23-s − 5·25-s − 0.405·29-s + 0.838·32-s − 9.99·37-s + 11.6·43-s − 25.4·44-s + 16.1·46-s + 12.1·50-s − 1.21·53-s + 0.984·58-s − 8.99·64-s + 14.5·67-s + 15.8·71-s + 24.3·74-s + 17.3·79-s − 28.2·86-s + 30.2·88-s − 25.9·92-s − 19.5·100-s + 2.96·106-s + ⋯ |
| L(s) = 1 | − 1.71·2-s + 1.95·4-s − 1.64·8-s − 1.96·11-s + 0.869·16-s + 3.37·22-s − 1.38·23-s − 25-s − 0.0752·29-s + 0.148·32-s − 1.64·37-s + 1.77·43-s − 3.83·44-s + 2.37·46-s + 1.71·50-s − 0.167·53-s + 0.129·58-s − 1.12·64-s + 1.78·67-s + 1.87·71-s + 2.82·74-s + 1.95·79-s − 3.04·86-s + 3.22·88-s − 2.70·92-s − 1.95·100-s + 0.288·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3984255184\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3984255184\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 2.43T + 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 6.50T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 6.62T + 23T^{2} \) |
| 29 | \( 1 + 0.405T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 9.99T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 1.21T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 14.5T + 67T^{2} \) |
| 71 | \( 1 - 15.8T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 17.3T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−8.593923494837991943613575736443, −7.947305638573905410103761133401, −7.66579795474104621922068901289, −6.75834524903670754498798019871, −5.84579341944282090208580470196, −5.07935548477648698207019034182, −3.76381707645977716330662344639, −2.52550220091993052855013240015, −1.96039604002130465297968248984, −0.46856514471369012862745197180,
0.46856514471369012862745197180, 1.96039604002130465297968248984, 2.52550220091993052855013240015, 3.76381707645977716330662344639, 5.07935548477648698207019034182, 5.84579341944282090208580470196, 6.75834524903670754498798019871, 7.66579795474104621922068901289, 7.947305638573905410103761133401, 8.593923494837991943613575736443
