| L(s) = 1 | + 3.34·3-s − 1.14·5-s − 1.14·7-s + 8.17·9-s + 3.14·11-s + 2.48·13-s − 3.83·15-s + 0.853·17-s − 5.66·19-s − 3.83·21-s + 23-s − 3.68·25-s + 17.3·27-s − 6.88·29-s + 8.32·31-s + 10.5·33-s + 1.31·35-s + 8.81·37-s + 8.32·39-s − 6.48·41-s − 2.97·43-s − 9.37·45-s + 2.94·47-s − 5.68·49-s + 2.85·51-s + 0.393·53-s − 3.60·55-s + ⋯ |
| L(s) = 1 | + 1.93·3-s − 0.512·5-s − 0.433·7-s + 2.72·9-s + 0.948·11-s + 0.690·13-s − 0.989·15-s + 0.207·17-s − 1.29·19-s − 0.836·21-s + 0.208·23-s − 0.737·25-s + 3.32·27-s − 1.27·29-s + 1.49·31-s + 1.83·33-s + 0.222·35-s + 1.44·37-s + 1.33·39-s − 1.01·41-s − 0.454·43-s − 1.39·45-s + 0.430·47-s − 0.812·49-s + 0.399·51-s + 0.0539·53-s − 0.486·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.758044285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.758044285\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 3.34T + 3T^{2} \) |
| 5 | \( 1 + 1.14T + 5T^{2} \) |
| 7 | \( 1 + 1.14T + 7T^{2} \) |
| 11 | \( 1 - 3.14T + 11T^{2} \) |
| 13 | \( 1 - 2.48T + 13T^{2} \) |
| 17 | \( 1 - 0.853T + 17T^{2} \) |
| 19 | \( 1 + 5.66T + 19T^{2} \) |
| 29 | \( 1 + 6.88T + 29T^{2} \) |
| 31 | \( 1 - 8.32T + 31T^{2} \) |
| 37 | \( 1 - 8.81T + 37T^{2} \) |
| 41 | \( 1 + 6.48T + 41T^{2} \) |
| 43 | \( 1 + 2.97T + 43T^{2} \) |
| 47 | \( 1 - 2.94T + 47T^{2} \) |
| 53 | \( 1 - 0.393T + 53T^{2} \) |
| 59 | \( 1 - 5.70T + 59T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 67 | \( 1 - 7.93T + 67T^{2} \) |
| 71 | \( 1 + 0.657T + 71T^{2} \) |
| 73 | \( 1 + 1.90T + 73T^{2} \) |
| 79 | \( 1 + 16.0T + 79T^{2} \) |
| 83 | \( 1 - 2.75T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−10.00003546005810425401297105932, −9.409651837768657222323137271995, −8.555178742934764515891742128175, −8.081479469100617876311578489248, −7.09963546854443705943606339220, −6.25494277948132076673266455188, −4.33109230713658439406333418438, −3.79903372904429536299855850406, −2.84693621223897119436587631338, −1.59402326884510172643041101605,
1.59402326884510172643041101605, 2.84693621223897119436587631338, 3.79903372904429536299855850406, 4.33109230713658439406333418438, 6.25494277948132076673266455188, 7.09963546854443705943606339220, 8.081479469100617876311578489248, 8.555178742934764515891742128175, 9.409651837768657222323137271995, 10.00003546005810425401297105932
