| L(s) = 1 | − 5.35·3-s + 20.7·5-s − 26.2·7-s + 1.71·9-s + 11.3·11-s + 82.2·13-s − 110.·15-s + 74.7·17-s − 49.4·19-s + 140.·21-s − 126.·23-s + 303.·25-s + 135.·27-s − 29·29-s + 169.·31-s − 60.6·33-s − 542.·35-s − 333.·37-s − 440.·39-s − 222.·41-s + 474.·43-s + 35.6·45-s + 49.3·47-s + 343.·49-s − 400.·51-s − 409.·53-s + 234.·55-s + ⋯ |
| L(s) = 1 | − 1.03·3-s + 1.85·5-s − 1.41·7-s + 0.0636·9-s + 0.310·11-s + 1.75·13-s − 1.90·15-s + 1.06·17-s − 0.596·19-s + 1.45·21-s − 1.14·23-s + 2.42·25-s + 0.965·27-s − 0.185·29-s + 0.980·31-s − 0.319·33-s − 2.62·35-s − 1.48·37-s − 1.80·39-s − 0.846·41-s + 1.68·43-s + 0.117·45-s + 0.153·47-s + 1.00·49-s − 1.09·51-s − 1.06·53-s + 0.574·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.915427217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.915427217\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 + 29T \) |
| good | 3 | \( 1 + 5.35T + 27T^{2} \) |
| 5 | \( 1 - 20.7T + 125T^{2} \) |
| 7 | \( 1 + 26.2T + 343T^{2} \) |
| 11 | \( 1 - 11.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 82.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 74.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 49.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 126.T + 1.21e4T^{2} \) |
| 31 | \( 1 - 169.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 333.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 222.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 474.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 49.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 409.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 281.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 407.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 296.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 483.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 152.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 657.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 9.66T + 5.71e5T^{2} \) |
| 89 | \( 1 + 647.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 540.T + 9.12e5T^{2} \) |
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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
−9.111122314176807623450138670267, −8.284608353819555347004452765368, −6.67775916871884704303949251869, −6.33936595921767010709648586230, −5.86786060308456725708424303804, −5.25996136340954709002446773341, −3.83376483195164506438699664419, −2.89956340018518848879029982117, −1.68185257868308230089828628717, −0.70360611634289038872795397778,
0.70360611634289038872795397778, 1.68185257868308230089828628717, 2.89956340018518848879029982117, 3.83376483195164506438699664419, 5.25996136340954709002446773341, 5.86786060308456725708424303804, 6.33936595921767010709648586230, 6.67775916871884704303949251869, 8.284608353819555347004452765368, 9.111122314176807623450138670267
