| L(s) = 1 | − 2.49·2-s + 4.21·4-s + 0.321·5-s − 5.52·8-s − 0.802·10-s − 2.79·11-s − 5.56·13-s + 5.34·16-s − 2.70·17-s + 5.07·19-s + 1.35·20-s + 6.97·22-s + 0.447·23-s − 4.89·25-s + 13.8·26-s − 7.86·29-s − 9.68·31-s − 2.27·32-s + 6.75·34-s − 2.85·37-s − 12.6·38-s − 1.77·40-s − 1.82·41-s + 4.04·43-s − 11.8·44-s − 1.11·46-s + 11.7·47-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 2.10·4-s + 0.143·5-s − 1.95·8-s − 0.253·10-s − 0.843·11-s − 1.54·13-s + 1.33·16-s − 0.656·17-s + 1.16·19-s + 0.303·20-s + 1.48·22-s + 0.0933·23-s − 0.979·25-s + 2.72·26-s − 1.45·29-s − 1.74·31-s − 0.401·32-s + 1.15·34-s − 0.469·37-s − 2.05·38-s − 0.281·40-s − 0.284·41-s + 0.616·43-s − 1.77·44-s − 0.164·46-s + 1.71·47-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
| \(L(1)\) |
\(\approx\) |
\(0.4400037878\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4400037878\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.49T + 2T^{2} \) |
| 5 | \( 1 - 0.321T + 5T^{2} \) |
| 11 | \( 1 + 2.79T + 11T^{2} \) |
| 13 | \( 1 + 5.56T + 13T^{2} \) |
| 17 | \( 1 + 2.70T + 17T^{2} \) |
| 19 | \( 1 - 5.07T + 19T^{2} \) |
| 23 | \( 1 - 0.447T + 23T^{2} \) |
| 29 | \( 1 + 7.86T + 29T^{2} \) |
| 31 | \( 1 + 9.68T + 31T^{2} \) |
| 37 | \( 1 + 2.85T + 37T^{2} \) |
| 41 | \( 1 + 1.82T + 41T^{2} \) |
| 43 | \( 1 - 4.04T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 59 | \( 1 - 2.08T + 59T^{2} \) |
| 61 | \( 1 - 8.56T + 61T^{2} \) |
| 67 | \( 1 + 9.07T + 67T^{2} \) |
| 71 | \( 1 - 5.92T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 - 7.26T + 79T^{2} \) |
| 83 | \( 1 - 2.64T + 83T^{2} \) |
| 89 | \( 1 + 5.04T + 89T^{2} \) |
| 97 | \( 1 - 18.0T + 97T^{2} \) |
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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
−8.981437272549779745564502594578, −7.84080194556275203581619081194, −7.43546752389140803963178635720, −7.00088078061519468635687467743, −5.74203894209406957669592928187, −5.15398893868856521222085941197, −3.74659924110615377053723485754, −2.46412247241484953920021444982, −1.99330980854685368311834766605, −0.49512820634704846400664473320,
0.49512820634704846400664473320, 1.99330980854685368311834766605, 2.46412247241484953920021444982, 3.74659924110615377053723485754, 5.15398893868856521222085941197, 5.74203894209406957669592928187, 7.00088078061519468635687467743, 7.43546752389140803963178635720, 7.84080194556275203581619081194, 8.981437272549779745564502594578
