L(s) = 1 | + 5·5-s + 8.37·7-s + 9.42·11-s + 43.8·13-s − 73.3·17-s + 54.5·19-s + 181.·23-s + 25·25-s − 147.·29-s + 83.2·31-s + 41.8·35-s − 219.·37-s + 220.·41-s + 223.·43-s + 254.·47-s − 272.·49-s − 394.·53-s + 47.1·55-s + 230.·59-s + 899.·61-s + 219.·65-s − 643.·67-s + 132.·71-s + 543.·73-s + 78.9·77-s − 390.·79-s − 434.·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.452·7-s + 0.258·11-s + 0.935·13-s − 1.04·17-s + 0.658·19-s + 1.64·23-s + 0.200·25-s − 0.941·29-s + 0.482·31-s + 0.202·35-s − 0.974·37-s + 0.839·41-s + 0.791·43-s + 0.790·47-s − 0.795·49-s − 1.02·53-s + 0.115·55-s + 0.508·59-s + 1.88·61-s + 0.418·65-s − 1.17·67-s + 0.221·71-s + 0.870·73-s + 0.116·77-s − 0.555·79-s − 0.574·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.840443009\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.840443009\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
good | 7 | \( 1 - 8.37T + 343T^{2} \) |
| 11 | \( 1 - 9.42T + 1.33e3T^{2} \) |
| 13 | \( 1 - 43.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 73.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 54.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 181.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 147.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 83.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 219.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 220.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 223.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 254.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 394.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 230.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 899.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 643.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 132.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 543.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 390.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 434.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 218.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.77e3T + 9.12e5T^{2} \) |
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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
−9.023093194580513344298690590663, −8.395847591468564404165095733259, −7.34649393035448434237161709599, −6.64694784429656375407450811644, −5.74671770727637576897352111440, −4.95746075524638350237215219267, −4.00334090685181457495653972027, −2.95830435385935775869479039249, −1.82851942618860751777351020890, −0.849962482988360793579130702457,
0.849962482988360793579130702457, 1.82851942618860751777351020890, 2.95830435385935775869479039249, 4.00334090685181457495653972027, 4.95746075524638350237215219267, 5.74671770727637576897352111440, 6.64694784429656375407450811644, 7.34649393035448434237161709599, 8.395847591468564404165095733259, 9.023093194580513344298690590663