| L(s) = 1 | − 4·5-s + 2·11-s − 5·13-s − 4·19-s + 23-s + 11·25-s + 3·29-s + 5·31-s + 4·37-s + 5·41-s + 4·43-s + 11·47-s − 8·55-s + 12·59-s + 6·61-s + 20·65-s + 16·67-s + 7·71-s − 7·73-s − 16·79-s − 8·83-s − 12·89-s + 16·95-s − 6·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 0.603·11-s − 1.38·13-s − 0.917·19-s + 0.208·23-s + 11/5·25-s + 0.557·29-s + 0.898·31-s + 0.657·37-s + 0.780·41-s + 0.609·43-s + 1.60·47-s − 1.07·55-s + 1.56·59-s + 0.768·61-s + 2.48·65-s + 1.95·67-s + 0.830·71-s − 0.819·73-s − 1.80·79-s − 0.878·83-s − 1.27·89-s + 1.64·95-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.807255286\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.807255286\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−12.99771834302407, −12.65801482665865, −12.30630033351837, −11.84297807128151, −11.42024242998783, −11.06513981007820, −10.47970635610826, −9.895504145714220, −9.505210621101659, −8.607683429987155, −8.580727473818485, −7.910470152825296, −7.461391580949494, −6.903104601182127, −6.755130397420573, −5.841707809440199, −5.285215859846969, −4.547303477865270, −4.163261889633123, −4.012306107779578, −3.012834989696982, −2.693341198880580, −1.962891891353296, −0.7830439858422918, −0.5694148037325393,
0.5694148037325393, 0.7830439858422918, 1.962891891353296, 2.693341198880580, 3.012834989696982, 4.012306107779578, 4.163261889633123, 4.547303477865270, 5.285215859846969, 5.841707809440199, 6.755130397420573, 6.903104601182127, 7.461391580949494, 7.910470152825296, 8.580727473818485, 8.607683429987155, 9.505210621101659, 9.895504145714220, 10.47970635610826, 11.06513981007820, 11.42024242998783, 11.84297807128151, 12.30630033351837, 12.65801482665865, 12.99771834302407
