| L(s) = 1 | + 2·2-s + 9.59·3-s + 4·4-s + 12.3·5-s + 19.1·6-s + 8·8-s + 65.1·9-s + 24.6·10-s + 3.20·11-s + 38.3·12-s + 90.5·13-s + 118.·15-s + 16·16-s − 39.3·17-s + 130.·18-s − 31.2·19-s + 49.2·20-s + 6.41·22-s + 23·23-s + 76.7·24-s + 26.7·25-s + 181.·26-s + 366.·27-s − 100.·29-s + 236.·30-s + 101.·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.84·3-s + 0.5·4-s + 1.10·5-s + 1.30·6-s + 0.353·8-s + 2.41·9-s + 0.779·10-s + 0.0878·11-s + 0.923·12-s + 1.93·13-s + 2.03·15-s + 0.250·16-s − 0.561·17-s + 1.70·18-s − 0.377·19-s + 0.550·20-s + 0.0621·22-s + 0.208·23-s + 0.653·24-s + 0.213·25-s + 1.36·26-s + 2.61·27-s − 0.646·29-s + 1.43·30-s + 0.589·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(10.53705691\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.53705691\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 - 9.59T + 27T^{2} \) |
| 5 | \( 1 - 12.3T + 125T^{2} \) |
| 11 | \( 1 - 3.20T + 1.33e3T^{2} \) |
| 13 | \( 1 - 90.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 39.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 31.2T + 6.85e3T^{2} \) |
| 29 | \( 1 + 100.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 101.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 276.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 428.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 189.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 461.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 532.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 535.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 61.3T + 2.26e5T^{2} \) |
| 67 | \( 1 - 324.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 339.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 493.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.02e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 967.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 141.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 533.T + 9.12e5T^{2} \) |
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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
−8.658504228116426241188208050436, −8.142064796805728310691649074972, −7.01430337073653529760959652011, −6.42439820140798828214486529306, −5.52550366664317034129135501708, −4.37281976961911260461538252467, −3.62907308970069579032992160957, −2.94604309621929523759574464492, −1.93114087996383096355514159695, −1.45668517518567201767053958295,
1.45668517518567201767053958295, 1.93114087996383096355514159695, 2.94604309621929523759574464492, 3.62907308970069579032992160957, 4.37281976961911260461538252467, 5.52550366664317034129135501708, 6.42439820140798828214486529306, 7.01430337073653529760959652011, 8.142064796805728310691649074972, 8.658504228116426241188208050436
