| L(s) = 1 | − 0.414·2-s − 1.82·4-s + 5-s + 2·7-s + 1.58·8-s − 0.414·10-s + 6.82·13-s − 0.828·14-s + 3·16-s + 1.17·17-s − 1.82·20-s − 2.82·23-s + 25-s − 2.82·26-s − 3.65·28-s + 7.65·29-s − 4.41·32-s − 0.485·34-s + 2·35-s + 3.65·37-s + 1.58·40-s + 6·41-s + 6·43-s + 1.17·46-s + 2.82·47-s − 3·49-s − 0.414·50-s + ⋯ |
| L(s) = 1 | − 0.292·2-s − 0.914·4-s + 0.447·5-s + 0.755·7-s + 0.560·8-s − 0.130·10-s + 1.89·13-s − 0.221·14-s + 0.750·16-s + 0.284·17-s − 0.408·20-s − 0.589·23-s + 0.200·25-s − 0.554·26-s − 0.691·28-s + 1.42·29-s − 0.780·32-s − 0.0832·34-s + 0.338·35-s + 0.601·37-s + 0.250·40-s + 0.937·41-s + 0.914·43-s + 0.172·46-s + 0.412·47-s − 0.428·49-s − 0.0585·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.965430476\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.965430476\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.414T + 2T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 13 | \( 1 - 6.82T + 13T^{2} \) |
| 17 | \( 1 - 1.17T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 - 7.65T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 3.65T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + 0.343T + 53T^{2} \) |
| 59 | \( 1 - 9.65T + 59T^{2} \) |
| 61 | \( 1 + 13.3T + 61T^{2} \) |
| 67 | \( 1 + 4.48T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 6.82T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + 9.31T + 89T^{2} \) |
| 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
−8.184634255285142865140303647380, −7.82935500034133719715214491057, −6.65921136390532920334326821949, −5.92011876838888545158631286331, −5.33720244303735127442092987412, −4.38992513739548570022790784955, −3.91424476464776678903641097137, −2.81908782308141833850290925488, −1.54984122971752724080285589921, −0.894211443976235578482143057558,
0.894211443976235578482143057558, 1.54984122971752724080285589921, 2.81908782308141833850290925488, 3.91424476464776678903641097137, 4.38992513739548570022790784955, 5.33720244303735127442092987412, 5.92011876838888545158631286331, 6.65921136390532920334326821949, 7.82935500034133719715214491057, 8.184634255285142865140303647380
