| L(s) = 1 | + 5.20·2-s + 4.90·3-s + 19.0·4-s − 5·5-s + 25.5·6-s + 57.6·8-s − 2.92·9-s − 26.0·10-s + 56.6·11-s + 93.6·12-s − 43.4·13-s − 24.5·15-s + 147.·16-s − 39.8·17-s − 15.2·18-s + 52.3·19-s − 95.3·20-s + 294.·22-s − 53.5·23-s + 282.·24-s + 25·25-s − 225.·26-s − 146.·27-s + 49.6·29-s − 127.·30-s − 73.7·31-s + 305.·32-s + ⋯ |
| L(s) = 1 | + 1.83·2-s + 0.944·3-s + 2.38·4-s − 0.447·5-s + 1.73·6-s + 2.54·8-s − 0.108·9-s − 0.822·10-s + 1.55·11-s + 2.25·12-s − 0.926·13-s − 0.422·15-s + 2.30·16-s − 0.568·17-s − 0.199·18-s + 0.632·19-s − 1.06·20-s + 2.85·22-s − 0.485·23-s + 2.40·24-s + 0.200·25-s − 1.70·26-s − 1.04·27-s + 0.317·29-s − 0.776·30-s − 0.427·31-s + 1.68·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.583177272\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.583177272\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 5.20T + 8T^{2} \) |
| 3 | \( 1 - 4.90T + 27T^{2} \) |
| 11 | \( 1 - 56.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 43.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 39.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 52.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 53.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 49.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 73.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 307.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 292.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 365.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 442.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 25.7T + 1.48e5T^{2} \) |
| 59 | \( 1 - 376.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 632.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 511.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 134.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 409.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 926.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 296.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 488.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 475.T + 9.12e5T^{2} \) |
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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
−11.83549446605195013228697741135, −11.32421077680168475243375764046, −9.770350853210174141911290498219, −8.562032722414532027584751576189, −7.32493716433312517421926999076, −6.49602010793319599771784426256, −5.17262302880242219585273802202, −4.02974833085977721617996502202, −3.27095166916644933919952310672, −2.03048246197587978143469051478,
2.03048246197587978143469051478, 3.27095166916644933919952310672, 4.02974833085977721617996502202, 5.17262302880242219585273802202, 6.49602010793319599771784426256, 7.32493716433312517421926999076, 8.562032722414532027584751576189, 9.770350853210174141911290498219, 11.32421077680168475243375764046, 11.83549446605195013228697741135
