| L(s) = 1 | + 0.645·2-s − 1.58·4-s − 1.34·5-s + 1.48·7-s − 2.31·8-s − 0.871·10-s + 0.690·11-s + 6.48·13-s + 0.961·14-s + 1.67·16-s + 5.34·17-s + 4.77·19-s + 2.13·20-s + 0.446·22-s + 5.85·23-s − 3.17·25-s + 4.18·26-s − 2.35·28-s − 8.79·29-s − 5.47·31-s + 5.70·32-s + 3.45·34-s − 2.01·35-s + 1.09·37-s + 3.08·38-s + 3.12·40-s + 7.41·41-s + ⋯ |
| L(s) = 1 | + 0.456·2-s − 0.791·4-s − 0.603·5-s + 0.563·7-s − 0.817·8-s − 0.275·10-s + 0.208·11-s + 1.79·13-s + 0.257·14-s + 0.418·16-s + 1.29·17-s + 1.09·19-s + 0.477·20-s + 0.0950·22-s + 1.22·23-s − 0.635·25-s + 0.821·26-s − 0.445·28-s − 1.63·29-s − 0.983·31-s + 1.00·32-s + 0.592·34-s − 0.339·35-s + 0.180·37-s + 0.500·38-s + 0.493·40-s + 1.15·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.535348290\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.535348290\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 0.645T + 2T^{2} \) |
| 5 | \( 1 + 1.34T + 5T^{2} \) |
| 7 | \( 1 - 1.48T + 7T^{2} \) |
| 11 | \( 1 - 0.690T + 11T^{2} \) |
| 13 | \( 1 - 6.48T + 13T^{2} \) |
| 17 | \( 1 - 5.34T + 17T^{2} \) |
| 19 | \( 1 - 4.77T + 19T^{2} \) |
| 23 | \( 1 - 5.85T + 23T^{2} \) |
| 29 | \( 1 + 8.79T + 29T^{2} \) |
| 31 | \( 1 + 5.47T + 31T^{2} \) |
| 37 | \( 1 - 1.09T + 37T^{2} \) |
| 41 | \( 1 - 7.41T + 41T^{2} \) |
| 43 | \( 1 + 5.83T + 43T^{2} \) |
| 47 | \( 1 - 3.84T + 47T^{2} \) |
| 53 | \( 1 - 4.02T + 53T^{2} \) |
| 59 | \( 1 - 6.76T + 59T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 + 4.01T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 + 1.70T + 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 - 1.10T + 97T^{2} \) |
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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
−11.12481425794042909649285705233, −9.787368631587663319903278245424, −8.976014824410118763377437149497, −8.159762645527857064148220773349, −7.34170906254370796228773198913, −5.85392596076079248716261027132, −5.23737508409146616759613007244, −3.91688341228181469496621522422, −3.41417722470120207877197022945, −1.15462171270501754712954674420,
1.15462171270501754712954674420, 3.41417722470120207877197022945, 3.91688341228181469496621522422, 5.23737508409146616759613007244, 5.85392596076079248716261027132, 7.34170906254370796228773198913, 8.159762645527857064148220773349, 8.976014824410118763377437149497, 9.787368631587663319903278245424, 11.12481425794042909649285705233
