| L(s) = 1 | − 2·2-s + 4·4-s + 5·5-s + 7·7-s − 8·8-s + 9·9-s − 10·10-s − 14·14-s + 16·16-s − 6·17-s − 18·18-s − 18·19-s + 20·20-s + 25·25-s + 28·28-s − 32·32-s + 12·34-s + 35·35-s + 36·36-s + 66·37-s + 36·38-s − 40·40-s + 54·43-s + 45·45-s − 66·47-s + 49·49-s − 50·50-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s + 9-s − 10-s − 14-s + 16-s − 0.352·17-s − 18-s − 0.947·19-s + 20-s + 25-s + 28-s − 32-s + 6/17·34-s + 35-s + 36-s + 1.78·37-s + 0.947·38-s − 40-s + 1.25·43-s + 45-s − 1.40·47-s + 49-s − 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.455600220\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.455600220\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 - p T \) |
| good | 3 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( ( 1 - p T )( 1 + p T ) \) |
| 17 | \( 1 + 6 T + p^{2} T^{2} \) |
| 19 | \( 1 + 18 T + p^{2} T^{2} \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( 1 - 66 T + p^{2} T^{2} \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( 1 - 54 T + p^{2} T^{2} \) |
| 47 | \( 1 + 66 T + p^{2} T^{2} \) |
| 53 | \( 1 - 34 T + p^{2} T^{2} \) |
| 59 | \( 1 - 62 T + p^{2} T^{2} \) |
| 61 | \( 1 + 102 T + p^{2} T^{2} \) |
| 67 | \( 1 - 6 T + p^{2} T^{2} \) |
| 71 | \( 1 + 138 T + p^{2} T^{2} \) |
| 73 | \( 1 - 106 T + p^{2} T^{2} \) |
| 79 | \( 1 + 122 T + p^{2} T^{2} \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( 1 + 166 T + p^{2} T^{2} \) |
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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
−11.28657185701545217225327253111, −10.55817122476733967579742850081, −9.742634321042644829177190959286, −8.864887491881860027207091949653, −7.87602859941207756757017084979, −6.86049168749656352500220402693, −5.85431971029810532762047573370, −4.46087373265398122140926508635, −2.39092848480770464480995683151, −1.33395589168590521694273672164,
1.33395589168590521694273672164, 2.39092848480770464480995683151, 4.46087373265398122140926508635, 5.85431971029810532762047573370, 6.86049168749656352500220402693, 7.87602859941207756757017084979, 8.864887491881860027207091949653, 9.742634321042644829177190959286, 10.55817122476733967579742850081, 11.28657185701545217225327253111
