L(s) = 1 | + 2-s − 2·3-s + 4-s + 5-s − 2·6-s + 7-s + 8-s + 9-s + 10-s − 3·11-s − 2·12-s + 14-s − 2·15-s + 16-s − 6·17-s + 18-s − 5·19-s + 20-s − 2·21-s − 3·22-s − 2·24-s + 25-s + 4·27-s + 28-s − 2·30-s + 4·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.447·5-s − 0.816·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.904·11-s − 0.577·12-s + 0.267·14-s − 0.516·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 1.14·19-s + 0.223·20-s − 0.436·21-s − 0.639·22-s − 0.408·24-s + 1/5·25-s + 0.769·27-s + 0.188·28-s − 0.365·30-s + 0.718·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.846041606675891741822150546782, −8.148942240498101627900797388626, −6.88015076394369594561483084866, −6.46291383741108850122834031179, −5.54289903870472654946419623087, −4.97407401915391721646567803800, −4.24042124100882380919634231977, −2.80685291157200836582402958028, −1.80312599843925486564133724136, 0,
1.80312599843925486564133724136, 2.80685291157200836582402958028, 4.24042124100882380919634231977, 4.97407401915391721646567803800, 5.54289903870472654946419623087, 6.46291383741108850122834031179, 6.88015076394369594561483084866, 8.148942240498101627900797388626, 8.846041606675891741822150546782