L(s) = 1 | − 1.73·2-s + 2·3-s − 5·4-s − 1.73·5-s − 3.46·6-s + 13.8·7-s + 22.5·8-s − 23·9-s + 2.99·10-s + 13.8·11-s − 10·12-s − 23.9·14-s − 3.46·15-s + 1.00·16-s − 117·17-s + 39.8·18-s − 114.·19-s + 8.66·20-s + 27.7·21-s − 23.9·22-s + 78·23-s + 45.0·24-s − 122·25-s − 100·27-s − 69.2·28-s − 141·29-s + 5.99·30-s + ⋯ |
L(s) = 1 | − 0.612·2-s + 0.384·3-s − 0.625·4-s − 0.154·5-s − 0.235·6-s + 0.748·7-s + 0.995·8-s − 0.851·9-s + 0.0948·10-s + 0.379·11-s − 0.240·12-s − 0.458·14-s − 0.0596·15-s + 0.0156·16-s − 1.66·17-s + 0.521·18-s − 1.38·19-s + 0.0968·20-s + 0.287·21-s − 0.232·22-s + 0.707·23-s + 0.383·24-s − 0.975·25-s − 0.712·27-s − 0.467·28-s − 0.902·29-s + 0.0365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
good | 2 | \( 1 + 1.73T + 8T^{2} \) |
| 3 | \( 1 - 2T + 27T^{2} \) |
| 5 | \( 1 + 1.73T + 125T^{2} \) |
| 7 | \( 1 - 13.8T + 343T^{2} \) |
| 11 | \( 1 - 13.8T + 1.33e3T^{2} \) |
| 17 | \( 1 + 117T + 4.91e3T^{2} \) |
| 19 | \( 1 + 114.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 78T + 1.21e4T^{2} \) |
| 29 | \( 1 + 141T + 2.43e4T^{2} \) |
| 31 | \( 1 - 155.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 143.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 271.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 104T + 7.95e4T^{2} \) |
| 47 | \( 1 + 301.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 93T + 1.48e5T^{2} \) |
| 59 | \( 1 - 284.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 145T + 2.26e5T^{2} \) |
| 67 | \( 1 - 786.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.05e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 458.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.27e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 789.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 976.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 200.T + 9.12e5T^{2} \) |
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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
−11.53739697140449124685952361142, −10.83784945296324992791265207365, −9.541246133834264939397636365582, −8.583769886731941558990405224738, −8.179860160026847185608533425906, −6.69044845874048300218112888800, −5.06699261552184947049811489027, −3.94632069183710927577760454264, −2.00053452054774119728275720965, 0,
2.00053452054774119728275720965, 3.94632069183710927577760454264, 5.06699261552184947049811489027, 6.69044845874048300218112888800, 8.179860160026847185608533425906, 8.583769886731941558990405224738, 9.541246133834264939397636365582, 10.83784945296324992791265207365, 11.53739697140449124685952361142