| L(s) = 1 | − 2.41·2-s + 3.82·4-s + 2.82·5-s + 2.82·7-s − 4.41·8-s − 6.82·10-s − 2·11-s − 6.82·14-s + 2.99·16-s + 3.65·17-s − 2.82·19-s + 10.8·20-s + 4.82·22-s + 4·23-s + 3.00·25-s + 10.8·28-s − 2·29-s + 6.82·31-s + 1.58·32-s − 8.82·34-s + 8.00·35-s − 3.65·37-s + 6.82·38-s − 12.4·40-s + 10.8·41-s + 9.65·43-s − 7.65·44-s + ⋯ |
| L(s) = 1 | − 1.70·2-s + 1.91·4-s + 1.26·5-s + 1.06·7-s − 1.56·8-s − 2.15·10-s − 0.603·11-s − 1.82·14-s + 0.749·16-s + 0.886·17-s − 0.648·19-s + 2.42·20-s + 1.02·22-s + 0.834·23-s + 0.600·25-s + 2.04·28-s − 0.371·29-s + 1.22·31-s + 0.280·32-s − 1.51·34-s + 1.35·35-s − 0.601·37-s + 1.10·38-s − 1.97·40-s + 1.69·41-s + 1.47·43-s − 1.15·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.105354371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.105354371\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.41T + 2T^{2} \) |
| 5 | \( 1 - 2.82T + 5T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 - 3.65T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 6.82T + 31T^{2} \) |
| 37 | \( 1 + 3.65T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 - 9.65T + 43T^{2} \) |
| 47 | \( 1 + 0.343T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 3.65T + 59T^{2} \) |
| 61 | \( 1 + 9.31T + 61T^{2} \) |
| 67 | \( 1 + 1.17T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 7.65T + 83T^{2} \) |
| 89 | \( 1 - 9.17T + 89T^{2} \) |
| 97 | \( 1 - 7.65T + 97T^{2} \) |
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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
−9.353210459151241721163483888227, −8.845823712072260614020394633224, −7.928715437054607046535333197668, −7.47639678854518691721782362895, −6.36468669123078707658828234681, −5.62536092260477728866504768471, −4.61732307188731810326957189766, −2.76494285350949537120585963927, −1.94348077187133837272790771925, −1.02332194949292890214877486170,
1.02332194949292890214877486170, 1.94348077187133837272790771925, 2.76494285350949537120585963927, 4.61732307188731810326957189766, 5.62536092260477728866504768471, 6.36468669123078707658828234681, 7.47639678854518691721782362895, 7.928715437054607046535333197668, 8.845823712072260614020394633224, 9.353210459151241721163483888227
