| L(s) = 1 | − 2-s + 3·3-s + 4-s − 2·5-s − 3·6-s − 7-s − 8-s + 6·9-s + 2·10-s + 3·12-s + 7·13-s + 14-s − 6·15-s + 16-s + 3·17-s − 6·18-s − 19-s − 2·20-s − 3·21-s + 3·23-s − 3·24-s − 25-s − 7·26-s + 9·27-s − 28-s − 29-s + 6·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s + 1/2·4-s − 0.894·5-s − 1.22·6-s − 0.377·7-s − 0.353·8-s + 2·9-s + 0.632·10-s + 0.866·12-s + 1.94·13-s + 0.267·14-s − 1.54·15-s + 1/4·16-s + 0.727·17-s − 1.41·18-s − 0.229·19-s − 0.447·20-s − 0.654·21-s + 0.625·23-s − 0.612·24-s − 1/5·25-s − 1.37·26-s + 1.73·27-s − 0.188·28-s − 0.185·29-s + 1.09·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.504666786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.504666786\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 15 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.335153455648747198612856227587, −8.007531852693460494662687929890, −7.09495767208138097339947380173, −6.57424023399750454601718585157, −5.42817979719352535445927398246, −4.01707230627126858076296675700, −3.62446344645114159933493861022, −3.00534977478020988175029198084, −1.92901216890802988280555468235, −0.941694133698868480128181644298,
0.941694133698868480128181644298, 1.92901216890802988280555468235, 3.00534977478020988175029198084, 3.62446344645114159933493861022, 4.01707230627126858076296675700, 5.42817979719352535445927398246, 6.57424023399750454601718585157, 7.09495767208138097339947380173, 8.007531852693460494662687929890, 8.335153455648747198612856227587
