| L(s) = 1 | + 1.45·3-s − 2.26·5-s − 7-s − 0.876·9-s + 5.41·11-s + 3.23·13-s − 3.30·15-s + 2.83·17-s + 4.32·19-s − 1.45·21-s − 6.99·23-s + 0.138·25-s − 5.64·27-s + 8.06·29-s − 9.18·31-s + 7.89·33-s + 2.26·35-s − 0.0469·37-s + 4.71·39-s − 41-s + 6.31·43-s + 1.98·45-s − 5.26·47-s + 49-s + 4.12·51-s + 6.43·53-s − 12.2·55-s + ⋯ |
| L(s) = 1 | + 0.841·3-s − 1.01·5-s − 0.377·7-s − 0.292·9-s + 1.63·11-s + 0.897·13-s − 0.852·15-s + 0.686·17-s + 0.991·19-s − 0.317·21-s − 1.45·23-s + 0.0277·25-s − 1.08·27-s + 1.49·29-s − 1.64·31-s + 1.37·33-s + 0.383·35-s − 0.00771·37-s + 0.755·39-s − 0.156·41-s + 0.962·43-s + 0.296·45-s − 0.767·47-s + 0.142·49-s + 0.577·51-s + 0.884·53-s − 1.65·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.214395883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.214395883\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - 1.45T + 3T^{2} \) |
| 5 | \( 1 + 2.26T + 5T^{2} \) |
| 11 | \( 1 - 5.41T + 11T^{2} \) |
| 13 | \( 1 - 3.23T + 13T^{2} \) |
| 17 | \( 1 - 2.83T + 17T^{2} \) |
| 19 | \( 1 - 4.32T + 19T^{2} \) |
| 23 | \( 1 + 6.99T + 23T^{2} \) |
| 29 | \( 1 - 8.06T + 29T^{2} \) |
| 31 | \( 1 + 9.18T + 31T^{2} \) |
| 37 | \( 1 + 0.0469T + 37T^{2} \) |
| 43 | \( 1 - 6.31T + 43T^{2} \) |
| 47 | \( 1 + 5.26T + 47T^{2} \) |
| 53 | \( 1 - 6.43T + 53T^{2} \) |
| 59 | \( 1 + 2.45T + 59T^{2} \) |
| 61 | \( 1 - 5.28T + 61T^{2} \) |
| 67 | \( 1 - 8.78T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 - 2.42T + 73T^{2} \) |
| 79 | \( 1 + 4.92T + 79T^{2} \) |
| 83 | \( 1 - 1.63T + 83T^{2} \) |
| 89 | \( 1 - 1.68T + 89T^{2} \) |
| 97 | \( 1 - 18.8T + 97T^{2} \) |
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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
−8.241534544966517313353157753208, −7.82003949875317942007844083934, −6.97542620895372150261079737844, −6.21768203846340710784897521364, −5.47302530982602688042200398996, −4.15551565292593555432080331426, −3.67794498006096815571857060898, −3.23405883500851434111647657086, −1.96711374591079486660037823942, −0.811121171767296233426365265668,
0.811121171767296233426365265668, 1.96711374591079486660037823942, 3.23405883500851434111647657086, 3.67794498006096815571857060898, 4.15551565292593555432080331426, 5.47302530982602688042200398996, 6.21768203846340710784897521364, 6.97542620895372150261079737844, 7.82003949875317942007844083934, 8.241534544966517313353157753208
