| L(s) = 1 | − 3.35·3-s + 0.564·5-s − 7-s + 8.23·9-s + 3.02·11-s + 5.91·13-s − 1.89·15-s + 2.56·17-s + 6.33·19-s + 3.35·21-s + 23-s − 4.68·25-s − 17.5·27-s − 5.23·29-s + 4.22·31-s − 10.1·33-s − 0.564·35-s − 5.17·37-s − 19.8·39-s + 0.660·41-s + 4.91·43-s + 4.64·45-s − 10.0·47-s + 49-s − 8.59·51-s + 6·53-s + 1.70·55-s + ⋯ |
| L(s) = 1 | − 1.93·3-s + 0.252·5-s − 0.377·7-s + 2.74·9-s + 0.911·11-s + 1.64·13-s − 0.488·15-s + 0.622·17-s + 1.45·19-s + 0.731·21-s + 0.208·23-s − 0.936·25-s − 3.37·27-s − 0.971·29-s + 0.758·31-s − 1.76·33-s − 0.0954·35-s − 0.850·37-s − 3.17·39-s + 0.103·41-s + 0.749·43-s + 0.692·45-s − 1.46·47-s + 0.142·49-s − 1.20·51-s + 0.824·53-s + 0.230·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.152009987\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.152009987\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 3.35T + 3T^{2} \) |
| 5 | \( 1 - 0.564T + 5T^{2} \) |
| 11 | \( 1 - 3.02T + 11T^{2} \) |
| 13 | \( 1 - 5.91T + 13T^{2} \) |
| 17 | \( 1 - 2.56T + 17T^{2} \) |
| 19 | \( 1 - 6.33T + 19T^{2} \) |
| 29 | \( 1 + 5.23T + 29T^{2} \) |
| 31 | \( 1 - 4.22T + 31T^{2} \) |
| 37 | \( 1 + 5.17T + 37T^{2} \) |
| 41 | \( 1 - 0.660T + 41T^{2} \) |
| 43 | \( 1 - 4.91T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 - 5.47T + 61T^{2} \) |
| 67 | \( 1 - 8.34T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 - 6.20T + 73T^{2} \) |
| 79 | \( 1 + 4.34T + 79T^{2} \) |
| 83 | \( 1 + 6.99T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
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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
−9.148516417488195888191976712903, −7.956145325354831615322861623556, −6.98534528909665767706945809509, −6.48403730597141432055741998294, −5.67641891466900595308130377037, −5.38777038398113381282228304521, −4.13142839498812836924461091219, −3.52762732653632995166956248325, −1.57124578382546931865264560519, −0.830987210983123064216723576889,
0.830987210983123064216723576889, 1.57124578382546931865264560519, 3.52762732653632995166956248325, 4.13142839498812836924461091219, 5.38777038398113381282228304521, 5.67641891466900595308130377037, 6.48403730597141432055741998294, 6.98534528909665767706945809509, 7.956145325354831615322861623556, 9.148516417488195888191976712903
