| L(s) = 1 | − 2.41·2-s − 2.82·3-s + 3.82·4-s − 5-s + 6.82·6-s + 2·7-s − 4.41·8-s + 5.00·9-s + 2.41·10-s − 10.8·12-s + 1.17·13-s − 4.82·14-s + 2.82·15-s + 2.99·16-s − 6.82·17-s − 12.0·18-s − 3.82·20-s − 5.65·21-s − 2.82·23-s + 12.4·24-s + 25-s − 2.82·26-s − 5.65·27-s + 7.65·28-s + 3.65·29-s − 6.82·30-s + 1.58·32-s + ⋯ |
| L(s) = 1 | − 1.70·2-s − 1.63·3-s + 1.91·4-s − 0.447·5-s + 2.78·6-s + 0.755·7-s − 1.56·8-s + 1.66·9-s + 0.763·10-s − 3.12·12-s + 0.324·13-s − 1.29·14-s + 0.730·15-s + 0.749·16-s − 1.65·17-s − 2.84·18-s − 0.856·20-s − 1.23·21-s − 0.589·23-s + 2.54·24-s + 0.200·25-s − 0.554·26-s − 1.08·27-s + 1.44·28-s + 0.679·29-s − 1.24·30-s + 0.280·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2854885744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2854885744\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 3 | \( 1 + 2.82T + 3T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 13 | \( 1 - 1.17T + 13T^{2} \) |
| 17 | \( 1 + 6.82T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 - 3.65T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 7.65T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 - 1.65T + 59T^{2} \) |
| 61 | \( 1 - 9.31T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 1.17T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 - 3.65T + 97T^{2} \) |
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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
−10.69392841136218057500068232333, −10.01616586818574353301921913163, −8.818366946591205571871160512807, −8.176291199733775922108628626945, −7.03664113219270685312609507394, −6.55607289928911078161638991098, −5.33960226149944408115322300453, −4.26236798415421457702648897318, −2.00202795573304983660140301308, −0.64500020425035326738338827898,
0.64500020425035326738338827898, 2.00202795573304983660140301308, 4.26236798415421457702648897318, 5.33960226149944408115322300453, 6.55607289928911078161638991098, 7.03664113219270685312609507394, 8.176291199733775922108628626945, 8.818366946591205571871160512807, 10.01616586818574353301921913163, 10.69392841136218057500068232333
