| L(s) = 1 | + 7.23·3-s + 16.5·5-s + 17.9·7-s + 25.4·9-s + 24.0·11-s − 13·13-s + 120.·15-s + 17.6·17-s + 137.·19-s + 130.·21-s + 20.8·23-s + 149.·25-s − 11.5·27-s − 80.8·29-s + 8.05·31-s + 174.·33-s + 297.·35-s + 37.7·37-s − 94.1·39-s − 475.·41-s − 258.·43-s + 421.·45-s − 149.·47-s − 20.3·49-s + 128.·51-s + 450.·53-s + 398.·55-s + ⋯ |
| L(s) = 1 | + 1.39·3-s + 1.48·5-s + 0.969·7-s + 0.940·9-s + 0.659·11-s − 0.277·13-s + 2.06·15-s + 0.252·17-s + 1.65·19-s + 1.35·21-s + 0.188·23-s + 1.19·25-s − 0.0822·27-s − 0.517·29-s + 0.0466·31-s + 0.918·33-s + 1.43·35-s + 0.167·37-s − 0.386·39-s − 1.80·41-s − 0.918·43-s + 1.39·45-s − 0.464·47-s − 0.0594·49-s + 0.351·51-s + 1.16·53-s + 0.977·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.281816703\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.281816703\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 3 | \( 1 - 7.23T + 27T^{2} \) |
| 5 | \( 1 - 16.5T + 125T^{2} \) |
| 7 | \( 1 - 17.9T + 343T^{2} \) |
| 11 | \( 1 - 24.0T + 1.33e3T^{2} \) |
| 17 | \( 1 - 17.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 137.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 20.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 80.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 8.05T + 2.97e4T^{2} \) |
| 37 | \( 1 - 37.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 475.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 258.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 149.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 450.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 176.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 594.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 315.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 810.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 23.2T + 3.89e5T^{2} \) |
| 79 | \( 1 + 345.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 544.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 859.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.34e3T + 9.12e5T^{2} \) |
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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
−8.956460556184688162573463505545, −8.401323261624088535282044838943, −7.52876852044828620558700319587, −6.75791838768241485749640393890, −5.58721960142398567202434101083, −5.00043526751227087641247639330, −3.72268105341069905738715012085, −2.83971335146882209047194311444, −1.90082335626922205524253020237, −1.31024199861230557703062827311,
1.31024199861230557703062827311, 1.90082335626922205524253020237, 2.83971335146882209047194311444, 3.72268105341069905738715012085, 5.00043526751227087641247639330, 5.58721960142398567202434101083, 6.75791838768241485749640393890, 7.52876852044828620558700319587, 8.401323261624088535282044838943, 8.956460556184688162573463505545
