| L(s) = 1 | + 2-s + 4-s + 5-s + 8-s + 10-s − 3·11-s + 16-s + 2·17-s − 2·19-s + 20-s − 3·22-s − 23-s + 25-s − 5·29-s + 5·31-s + 32-s + 2·34-s − 3·37-s − 2·38-s + 40-s − 2·41-s − 9·43-s − 3·44-s − 46-s − 7·47-s − 7·49-s + 50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s − 0.904·11-s + 1/4·16-s + 0.485·17-s − 0.458·19-s + 0.223·20-s − 0.639·22-s − 0.208·23-s + 1/5·25-s − 0.928·29-s + 0.898·31-s + 0.176·32-s + 0.342·34-s − 0.493·37-s − 0.324·38-s + 0.158·40-s − 0.312·41-s − 1.37·43-s − 0.452·44-s − 0.147·46-s − 1.02·47-s − 49-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 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
−16.34245096615096, −15.61383507088071, −15.12430990541607, −14.69361520535538, −14.00999150061480, −13.39813021309265, −13.19458453695533, −12.41942348335889, −11.99787651461567, −11.24775729146440, −10.73937001307726, −10.06363008128196, −9.725338388230163, −8.795616124071112, −8.116580739919211, −7.675077723550279, −6.730056866717167, −6.415591266384415, −5.390582451662444, −5.270597230549859, −4.383511837576387, −3.597398881820213, −2.905240166629896, −2.174079642360457, −1.387493706979039, 0,
1.387493706979039, 2.174079642360457, 2.905240166629896, 3.597398881820213, 4.383511837576387, 5.270597230549859, 5.390582451662444, 6.415591266384415, 6.730056866717167, 7.675077723550279, 8.116580739919211, 8.795616124071112, 9.725338388230163, 10.06363008128196, 10.73937001307726, 11.24775729146440, 11.99787651461567, 12.41942348335889, 13.19458453695533, 13.39813021309265, 14.00999150061480, 14.69361520535538, 15.12430990541607, 15.61383507088071, 16.34245096615096
