| L(s) = 1 | + 3-s − 7-s + 9-s − 4·11-s + 6·13-s − 2·17-s + 6·19-s − 21-s − 2·23-s + 27-s + 6·29-s − 2·31-s − 4·33-s + 4·37-s + 6·39-s + 8·41-s + 4·43-s − 4·47-s + 49-s − 2·51-s − 6·53-s + 6·57-s + 4·59-s + 14·61-s − 63-s − 4·67-s − 2·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 1.66·13-s − 0.485·17-s + 1.37·19-s − 0.218·21-s − 0.417·23-s + 0.192·27-s + 1.11·29-s − 0.359·31-s − 0.696·33-s + 0.657·37-s + 0.960·39-s + 1.24·41-s + 0.609·43-s − 0.583·47-s + 1/7·49-s − 0.280·51-s − 0.824·53-s + 0.794·57-s + 0.520·59-s + 1.79·61-s − 0.125·63-s − 0.488·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.155871180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.155871180\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 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
−9.116671878067058228668736642197, −8.217274976762957722158366546561, −7.79475966786471025761067987992, −6.77238615811618901030894496151, −5.97716076552890173486392763311, −5.12771157921464522957188091436, −4.03694670278567160946220010364, −3.22455670944000123351362763963, −2.37205064712529596769921999280, −0.973469207532226085894380069823,
0.973469207532226085894380069823, 2.37205064712529596769921999280, 3.22455670944000123351362763963, 4.03694670278567160946220010364, 5.12771157921464522957188091436, 5.97716076552890173486392763311, 6.77238615811618901030894496151, 7.79475966786471025761067987992, 8.217274976762957722158366546561, 9.116671878067058228668736642197
