L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s + 7-s + 8-s + 9-s + 3·11-s + 2·12-s + 14-s + 16-s + 6·17-s + 18-s + 5·19-s + 2·21-s + 3·22-s + 2·24-s − 4·27-s + 28-s − 4·31-s + 32-s + 6·33-s + 6·34-s + 36-s − 11·37-s + 5·38-s + 6·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.904·11-s + 0.577·12-s + 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 1.14·19-s + 0.436·21-s + 0.639·22-s + 0.408·24-s − 0.769·27-s + 0.188·28-s − 0.718·31-s + 0.176·32-s + 1.04·33-s + 1.02·34-s + 1/6·36-s − 1.80·37-s + 0.811·38-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.177360370\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.177360370\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 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
−7.55119083898535296527327534801, −7.38501029481137307214552433597, −6.40002054087471338583238047210, −5.51612784699853738472716566786, −5.09586862993195283589071991923, −3.88804478864690621149670586386, −3.59288761584868845491908675618, −2.85410512844210466244914788995, −1.93983097637206577967906349238, −1.12433009973949239496396988690,
1.12433009973949239496396988690, 1.93983097637206577967906349238, 2.85410512844210466244914788995, 3.59288761584868845491908675618, 3.88804478864690621149670586386, 5.09586862993195283589071991923, 5.51612784699853738472716566786, 6.40002054087471338583238047210, 7.38501029481137307214552433597, 7.55119083898535296527327534801