| L(s) = 1 | − 2·3-s − 5-s + 9-s + 4·11-s + 13-s + 2·15-s − 6·19-s + 2·23-s + 25-s + 4·27-s + 6·29-s − 8·31-s − 8·33-s − 6·37-s − 2·39-s + 8·41-s − 4·43-s − 45-s − 8·47-s − 4·55-s + 12·57-s − 10·59-s + 14·61-s − 65-s + 4·67-s − 4·69-s − 6·71-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s + 0.277·13-s + 0.516·15-s − 1.37·19-s + 0.417·23-s + 1/5·25-s + 0.769·27-s + 1.11·29-s − 1.43·31-s − 1.39·33-s − 0.986·37-s − 0.320·39-s + 1.24·41-s − 0.609·43-s − 0.149·45-s − 1.16·47-s − 0.539·55-s + 1.58·57-s − 1.30·59-s + 1.79·61-s − 0.124·65-s + 0.488·67-s − 0.481·69-s − 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8374087866\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8374087866\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.55205203404307, −14.14728312083822, −13.36004057912262, −12.64374092140573, −12.54519979351796, −11.79789102275586, −11.41461300449802, −11.07298421361129, −10.48254969020367, −10.04347005681328, −9.208135950813479, −8.708803014110645, −8.380492358909108, −7.486131086364004, −6.878860224544844, −6.504914045383443, −6.006027142016210, −5.409483309550009, −4.698312037623754, −4.273996103909836, −3.612381136644908, −2.932655381679546, −1.915676984547539, −1.226370190183740, −0.3762068967442344,
0.3762068967442344, 1.226370190183740, 1.915676984547539, 2.932655381679546, 3.612381136644908, 4.273996103909836, 4.698312037623754, 5.409483309550009, 6.006027142016210, 6.504914045383443, 6.878860224544844, 7.486131086364004, 8.380492358909108, 8.708803014110645, 9.208135950813479, 10.04347005681328, 10.48254969020367, 11.07298421361129, 11.41461300449802, 11.79789102275586, 12.54519979351796, 12.64374092140573, 13.36004057912262, 14.14728312083822, 14.55205203404307
