| L(s) = 1 | + 2.48·2-s − 1.82·4-s − 13.9·5-s − 7·7-s − 24.4·8-s − 34.5·10-s + 11·11-s − 53.9·13-s − 17.3·14-s − 46.0·16-s + 21.8·17-s + 73.6·19-s + 25.4·20-s + 27.3·22-s + 16.7·23-s + 68.8·25-s − 134.·26-s + 12.7·28-s − 153.·29-s + 224.·31-s + 80.9·32-s + 54.3·34-s + 97.4·35-s + 369.·37-s + 183.·38-s + 339.·40-s + 61.3·41-s + ⋯ |
| L(s) = 1 | + 0.878·2-s − 0.228·4-s − 1.24·5-s − 0.377·7-s − 1.07·8-s − 1.09·10-s + 0.301·11-s − 1.15·13-s − 0.332·14-s − 0.719·16-s + 0.312·17-s + 0.889·19-s + 0.284·20-s + 0.264·22-s + 0.151·23-s + 0.550·25-s − 1.01·26-s + 0.0863·28-s − 0.985·29-s + 1.29·31-s + 0.447·32-s + 0.274·34-s + 0.470·35-s + 1.64·37-s + 0.781·38-s + 1.34·40-s + 0.233·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.404022928\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.404022928\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 2.48T + 8T^{2} \) |
| 5 | \( 1 + 13.9T + 125T^{2} \) |
| 13 | \( 1 + 53.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 21.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 73.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 16.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 153.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 224.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 369.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 61.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 37.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 279.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 235.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 19.1T + 2.05e5T^{2} \) |
| 61 | \( 1 - 713.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 198.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 76.3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 984.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 697.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 388.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 33.0T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.38e3T + 9.12e5T^{2} \) |
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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
−9.901091105129874075502151016465, −9.330517122929462443318468332601, −8.161767374240959976915358803518, −7.45388282200177387094071988213, −6.41191127550993805991526774520, −5.29899489057506918211344072311, −4.45858094389607168367687061669, −3.65520268371966552563177240300, −2.76063192984768842385388783131, −0.58416626045651210375072582158,
0.58416626045651210375072582158, 2.76063192984768842385388783131, 3.65520268371966552563177240300, 4.45858094389607168367687061669, 5.29899489057506918211344072311, 6.41191127550993805991526774520, 7.45388282200177387094071988213, 8.161767374240959976915358803518, 9.330517122929462443318468332601, 9.901091105129874075502151016465
