| L(s) = 1 | − 3.82·3-s − 3.68·5-s + 23.1·7-s − 12.3·9-s − 16.6·11-s − 87.9·13-s + 14.0·15-s − 58.1·17-s − 102.·19-s − 88.5·21-s + 111.·23-s − 111.·25-s + 150.·27-s + 29·29-s − 146.·31-s + 63.7·33-s − 85.3·35-s + 77.3·37-s + 336.·39-s + 245.·41-s − 369.·43-s + 45.6·45-s + 3.90·47-s + 193.·49-s + 222.·51-s − 355.·53-s + 61.4·55-s + ⋯ |
| L(s) = 1 | − 0.735·3-s − 0.329·5-s + 1.25·7-s − 0.458·9-s − 0.457·11-s − 1.87·13-s + 0.242·15-s − 0.829·17-s − 1.23·19-s − 0.920·21-s + 1.01·23-s − 0.891·25-s + 1.07·27-s + 0.185·29-s − 0.847·31-s + 0.336·33-s − 0.412·35-s + 0.343·37-s + 1.38·39-s + 0.935·41-s − 1.31·43-s + 0.151·45-s + 0.0121·47-s + 0.564·49-s + 0.610·51-s − 0.922·53-s + 0.150·55-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
| \(L(2)\) |
\(\approx\) |
\(0.6282488729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6282488729\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 3.82T + 27T^{2} \) |
| 5 | \( 1 + 3.68T + 125T^{2} \) |
| 7 | \( 1 - 23.1T + 343T^{2} \) |
| 11 | \( 1 + 16.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 87.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 58.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 102.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 111.T + 1.21e4T^{2} \) |
| 31 | \( 1 + 146.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 77.3T + 5.06e4T^{2} \) |
| 41 | \( 1 - 245.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 369.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 3.90T + 1.03e5T^{2} \) |
| 53 | \( 1 + 355.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 223.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 308.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 548.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 121.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 228.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 665.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.41e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 955.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 598.T + 9.12e5T^{2} \) |
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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
−8.734238678100278130617933219289, −8.074655754601552015529396605680, −7.32796255336788021079890197010, −6.53332499483654270068715178685, −5.42056420932711594827198346386, −4.91918505203119311754376027250, −4.23773456362551392900354839712, −2.71830106858783285734144539110, −1.91352296387446365630176602713, −0.36626095455036408436420138306,
0.36626095455036408436420138306, 1.91352296387446365630176602713, 2.71830106858783285734144539110, 4.23773456362551392900354839712, 4.91918505203119311754376027250, 5.42056420932711594827198346386, 6.53332499483654270068715178685, 7.32796255336788021079890197010, 8.074655754601552015529396605680, 8.734238678100278130617933219289
