| L(s) = 1 | + 2.82·2-s − 0.0423·4-s + 3.41·5-s + 13.3·7-s − 22.6·8-s + 9.62·10-s − 35.4·11-s + 37.6·14-s − 63.6·16-s − 69.6·17-s + 12.4·19-s − 0.144·20-s − 100.·22-s + 126.·23-s − 113.·25-s − 0.565·28-s + 179.·29-s + 255.·31-s + 1.91·32-s − 196.·34-s + 45.5·35-s + 207.·37-s + 34.9·38-s − 77.3·40-s + 117.·41-s + 553.·43-s + 1.50·44-s + ⋯ |
| L(s) = 1 | + 0.997·2-s − 0.00529·4-s + 0.305·5-s + 0.720·7-s − 1.00·8-s + 0.304·10-s − 0.971·11-s + 0.718·14-s − 0.994·16-s − 0.993·17-s + 0.149·19-s − 0.00161·20-s − 0.969·22-s + 1.14·23-s − 0.906·25-s − 0.00381·28-s + 1.14·29-s + 1.48·31-s + 0.0105·32-s − 0.991·34-s + 0.219·35-s + 0.920·37-s + 0.149·38-s − 0.305·40-s + 0.448·41-s + 1.96·43-s + 0.00514·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.205205676\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.205205676\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.82T + 8T^{2} \) |
| 5 | \( 1 - 3.41T + 125T^{2} \) |
| 7 | \( 1 - 13.3T + 343T^{2} \) |
| 11 | \( 1 + 35.4T + 1.33e3T^{2} \) |
| 17 | \( 1 + 69.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 12.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 126.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 179.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 255.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 207.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 117.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 553.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 62.9T + 1.03e5T^{2} \) |
| 53 | \( 1 - 147.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 274.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 603.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 741.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 572.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 26.7T + 3.89e5T^{2} \) |
| 79 | \( 1 + 207.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.03e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.22e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.79e3T + 9.12e5T^{2} \) |
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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
−9.034865668038880850075643169609, −8.320422851830990062206249012866, −7.45665375501435955986726546746, −6.35735214730331143418666075600, −5.68199262465210473002767778582, −4.73620193770866623079235003372, −4.39572891634423934736538918205, −3.02230103656192380074740108803, −2.30726240557148487824241600044, −0.74351249167655555242503893070,
0.74351249167655555242503893070, 2.30726240557148487824241600044, 3.02230103656192380074740108803, 4.39572891634423934736538918205, 4.73620193770866623079235003372, 5.68199262465210473002767778582, 6.35735214730331143418666075600, 7.45665375501435955986726546746, 8.320422851830990062206249012866, 9.034865668038880850075643169609
