| L(s) = 1 | + 2.67·2-s + 1.01·3-s + 5.16·4-s + 1.61·5-s + 2.71·6-s − 4.48·7-s + 8.47·8-s − 1.96·9-s + 4.32·10-s − 11-s + 5.24·12-s + 0.368·13-s − 12.0·14-s + 1.63·15-s + 12.3·16-s + 1.42·17-s − 5.27·18-s + 1.32·19-s + 8.33·20-s − 4.55·21-s − 2.67·22-s + 8.55·23-s + 8.60·24-s − 2.39·25-s + 0.986·26-s − 5.04·27-s − 23.1·28-s + ⋯ |
| L(s) = 1 | + 1.89·2-s + 0.586·3-s + 2.58·4-s + 0.721·5-s + 1.10·6-s − 1.69·7-s + 2.99·8-s − 0.656·9-s + 1.36·10-s − 0.301·11-s + 1.51·12-s + 0.102·13-s − 3.21·14-s + 0.423·15-s + 3.08·16-s + 0.345·17-s − 1.24·18-s + 0.303·19-s + 1.86·20-s − 0.994·21-s − 0.570·22-s + 1.78·23-s + 1.75·24-s − 0.479·25-s + 0.193·26-s − 0.971·27-s − 4.38·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.042157612\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.042157612\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 2.67T + 2T^{2} \) |
| 3 | \( 1 - 1.01T + 3T^{2} \) |
| 5 | \( 1 - 1.61T + 5T^{2} \) |
| 7 | \( 1 + 4.48T + 7T^{2} \) |
| 13 | \( 1 - 0.368T + 13T^{2} \) |
| 17 | \( 1 - 1.42T + 17T^{2} \) |
| 19 | \( 1 - 1.32T + 19T^{2} \) |
| 23 | \( 1 - 8.55T + 23T^{2} \) |
| 29 | \( 1 + 2.21T + 29T^{2} \) |
| 31 | \( 1 + 8.44T + 31T^{2} \) |
| 37 | \( 1 + 1.07T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 + 6.14T + 43T^{2} \) |
| 47 | \( 1 - 5.43T + 47T^{2} \) |
| 53 | \( 1 - 7.19T + 53T^{2} \) |
| 59 | \( 1 - 0.842T + 59T^{2} \) |
| 67 | \( 1 + 8.44T + 67T^{2} \) |
| 71 | \( 1 - 9.05T + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 + 9.56T + 79T^{2} \) |
| 83 | \( 1 - 8.83T + 83T^{2} \) |
| 89 | \( 1 + 2.26T + 89T^{2} \) |
| 97 | \( 1 - 4.56T + 97T^{2} \) |
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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
−10.68711556388485004004562523569, −9.784632752033450708755055140731, −8.884767262070476446612651186098, −7.38601321177293697318830505952, −6.65573006849413110910836284262, −5.77482888594880890640426885630, −5.21187817945823384795479578341, −3.58012107085709014060942873410, −3.18812250072104696556571030334, −2.17755453656671185848824121201,
2.17755453656671185848824121201, 3.18812250072104696556571030334, 3.58012107085709014060942873410, 5.21187817945823384795479578341, 5.77482888594880890640426885630, 6.65573006849413110910836284262, 7.38601321177293697318830505952, 8.884767262070476446612651186098, 9.784632752033450708755055140731, 10.68711556388485004004562523569
