| L(s) = 1 | + 0.706·2-s − 0.343·3-s − 1.50·4-s − 2.37·5-s − 0.242·6-s + 4.91·7-s − 2.47·8-s − 2.88·9-s − 1.68·10-s − 1.70·11-s + 0.514·12-s − 3.97·13-s + 3.47·14-s + 0.815·15-s + 1.25·16-s + 2.74·17-s − 2.03·18-s + 0.965·19-s + 3.56·20-s − 1.68·21-s − 1.20·22-s + 6.75·23-s + 0.848·24-s + 0.655·25-s − 2.80·26-s + 2.01·27-s − 7.37·28-s + ⋯ |
| L(s) = 1 | + 0.499·2-s − 0.198·3-s − 0.750·4-s − 1.06·5-s − 0.0989·6-s + 1.85·7-s − 0.874·8-s − 0.960·9-s − 0.531·10-s − 0.513·11-s + 0.148·12-s − 1.10·13-s + 0.927·14-s + 0.210·15-s + 0.313·16-s + 0.666·17-s − 0.480·18-s + 0.221·19-s + 0.797·20-s − 0.367·21-s − 0.256·22-s + 1.40·23-s + 0.173·24-s + 0.131·25-s − 0.550·26-s + 0.388·27-s − 1.39·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1681 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1681 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.267403633\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.267403633\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 41 | \( 1 \) |
| good | 2 | \( 1 - 0.706T + 2T^{2} \) |
| 3 | \( 1 + 0.343T + 3T^{2} \) |
| 5 | \( 1 + 2.37T + 5T^{2} \) |
| 7 | \( 1 - 4.91T + 7T^{2} \) |
| 11 | \( 1 + 1.70T + 11T^{2} \) |
| 13 | \( 1 + 3.97T + 13T^{2} \) |
| 17 | \( 1 - 2.74T + 17T^{2} \) |
| 19 | \( 1 - 0.965T + 19T^{2} \) |
| 23 | \( 1 - 6.75T + 23T^{2} \) |
| 29 | \( 1 + 0.507T + 29T^{2} \) |
| 31 | \( 1 + 0.590T + 31T^{2} \) |
| 37 | \( 1 - 4.25T + 37T^{2} \) |
| 43 | \( 1 + 5.38T + 43T^{2} \) |
| 47 | \( 1 - 5.02T + 47T^{2} \) |
| 53 | \( 1 - 7.62T + 53T^{2} \) |
| 59 | \( 1 - 8.42T + 59T^{2} \) |
| 61 | \( 1 - 0.695T + 61T^{2} \) |
| 67 | \( 1 + 4.08T + 67T^{2} \) |
| 71 | \( 1 - 7.27T + 71T^{2} \) |
| 73 | \( 1 - 9.72T + 73T^{2} \) |
| 79 | \( 1 + 8.70T + 79T^{2} \) |
| 83 | \( 1 - 4.81T + 83T^{2} \) |
| 89 | \( 1 - 3.68T + 89T^{2} \) |
| 97 | \( 1 - 4.08T + 97T^{2} \) |
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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
−9.132670393524799568707533590138, −8.385166314958373231350912281911, −7.895475460867463791707052154644, −7.20230801861215937482537833397, −5.64122452591575976852488265196, −5.09550099742623633195550351446, −4.59336261805630862145964818839, −3.57740384991458788598395170013, −2.50904614597606876600846507198, −0.73123718787346015211742029394,
0.73123718787346015211742029394, 2.50904614597606876600846507198, 3.57740384991458788598395170013, 4.59336261805630862145964818839, 5.09550099742623633195550351446, 5.64122452591575976852488265196, 7.20230801861215937482537833397, 7.895475460867463791707052154644, 8.385166314958373231350912281911, 9.132670393524799568707533590138
