L(s) = 1 | + 2.17·2-s − 1.70·3-s + 2.70·4-s − 1.63·5-s − 3.70·6-s − 0.460·7-s + 1.53·8-s − 0.0783·9-s − 3.53·10-s + 6.17·11-s − 4.63·12-s − 4.07·13-s − 14-s + 2.78·15-s − 2.07·16-s + 0.630·17-s − 0.170·18-s + 7.12·19-s − 4.41·20-s + 0.787·21-s + 13.3·22-s − 0.170·23-s − 2.63·24-s − 2.34·25-s − 8.85·26-s + 5.26·27-s − 1.24·28-s + ⋯ |
L(s) = 1 | + 1.53·2-s − 0.986·3-s + 1.35·4-s − 0.729·5-s − 1.51·6-s − 0.174·7-s + 0.544·8-s − 0.0261·9-s − 1.11·10-s + 1.86·11-s − 1.33·12-s − 1.13·13-s − 0.267·14-s + 0.719·15-s − 0.519·16-s + 0.153·17-s − 0.0400·18-s + 1.63·19-s − 0.988·20-s + 0.171·21-s + 2.85·22-s − 0.0354·23-s − 0.537·24-s − 0.468·25-s − 1.73·26-s + 1.01·27-s − 0.235·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.259518786\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.259518786\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 61 | \( 1 - T \) |
good | 2 | \( 1 - 2.17T + 2T^{2} \) |
| 3 | \( 1 + 1.70T + 3T^{2} \) |
| 5 | \( 1 + 1.63T + 5T^{2} \) |
| 7 | \( 1 + 0.460T + 7T^{2} \) |
| 11 | \( 1 - 6.17T + 11T^{2} \) |
| 13 | \( 1 + 4.07T + 13T^{2} \) |
| 17 | \( 1 - 0.630T + 17T^{2} \) |
| 19 | \( 1 - 7.12T + 19T^{2} \) |
| 23 | \( 1 + 0.170T + 23T^{2} \) |
| 29 | \( 1 - 2.63T + 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 - 5.12T + 37T^{2} \) |
| 41 | \( 1 - 3.15T + 41T^{2} \) |
| 43 | \( 1 + 3.36T + 43T^{2} \) |
| 47 | \( 1 - 0.183T + 47T^{2} \) |
| 53 | \( 1 + 4.34T + 53T^{2} \) |
| 59 | \( 1 - 1.78T + 59T^{2} \) |
| 67 | \( 1 + 8.55T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 + 0.552T + 73T^{2} \) |
| 79 | \( 1 + 7.00T + 79T^{2} \) |
| 83 | \( 1 - 6.83T + 83T^{2} \) |
| 89 | \( 1 - 4.49T + 89T^{2} \) |
| 97 | \( 1 - 8.52T + 97T^{2} \) |
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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
−14.74188915955925889551267731368, −14.12993169800459317769318799095, −12.57870170637764794925473964344, −11.79827064229453738461075449865, −11.39047186786774567963303815080, −9.427904778574026366474308979743, −7.24047512489062929994710960697, −6.07948700598410505477793456601, −4.87962610121038899096096952870, −3.55212990466223432450231080247,
3.55212990466223432450231080247, 4.87962610121038899096096952870, 6.07948700598410505477793456601, 7.24047512489062929994710960697, 9.427904778574026366474308979743, 11.39047186786774567963303815080, 11.79827064229453738461075449865, 12.57870170637764794925473964344, 14.12993169800459317769318799095, 14.74188915955925889551267731368