L(s) = 1 | − 3-s + 7-s + 9-s − 11-s + 4·13-s − 3·17-s + 5·19-s − 21-s − 3·23-s − 27-s − 6·29-s + 8·31-s + 33-s + 7·37-s − 4·39-s + 9·41-s − 8·43-s + 3·47-s − 6·49-s + 3·51-s − 6·53-s − 5·57-s + 3·59-s + 14·61-s + 63-s − 2·67-s + 3·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s − 0.727·17-s + 1.14·19-s − 0.218·21-s − 0.625·23-s − 0.192·27-s − 1.11·29-s + 1.43·31-s + 0.174·33-s + 1.15·37-s − 0.640·39-s + 1.40·41-s − 1.21·43-s + 0.437·47-s − 6/7·49-s + 0.420·51-s − 0.824·53-s − 0.662·57-s + 0.390·59-s + 1.79·61-s + 0.125·63-s − 0.244·67-s + 0.361·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.605408394\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.605408394\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−8.533366363768036665894131485740, −7.907399722356293446239967589922, −7.14222300682144100020080834629, −6.22559717910638491456369263143, −5.72795927912524235626537501289, −4.80227563736251391425127187622, −4.09061939154891115410203564799, −3.08417434889935858016967485118, −1.89583451694284492613556111631, −0.802045474788085516711551537887,
0.802045474788085516711551537887, 1.89583451694284492613556111631, 3.08417434889935858016967485118, 4.09061939154891115410203564799, 4.80227563736251391425127187622, 5.72795927912524235626537501289, 6.22559717910638491456369263143, 7.14222300682144100020080834629, 7.907399722356293446239967589922, 8.533366363768036665894131485740