| L(s) = 1 | − 2.74·2-s − 3-s + 5.53·4-s + 1.03·5-s + 2.74·6-s + 3.45·7-s − 9.69·8-s + 9-s − 2.84·10-s − 1.50·11-s − 5.53·12-s + 0.989·13-s − 9.49·14-s − 1.03·15-s + 15.5·16-s + 17-s − 2.74·18-s + 6.75·19-s + 5.73·20-s − 3.45·21-s + 4.13·22-s − 1.90·23-s + 9.69·24-s − 3.92·25-s − 2.71·26-s − 27-s + 19.1·28-s + ⋯ |
| L(s) = 1 | − 1.94·2-s − 0.577·3-s + 2.76·4-s + 0.463·5-s + 1.12·6-s + 1.30·7-s − 3.42·8-s + 0.333·9-s − 0.900·10-s − 0.453·11-s − 1.59·12-s + 0.274·13-s − 2.53·14-s − 0.267·15-s + 3.88·16-s + 0.242·17-s − 0.646·18-s + 1.55·19-s + 1.28·20-s − 0.754·21-s + 0.880·22-s − 0.397·23-s + 1.97·24-s − 0.784·25-s − 0.532·26-s − 0.192·27-s + 3.61·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8321744314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8321744314\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 + 2.74T + 2T^{2} \) |
| 5 | \( 1 - 1.03T + 5T^{2} \) |
| 7 | \( 1 - 3.45T + 7T^{2} \) |
| 11 | \( 1 + 1.50T + 11T^{2} \) |
| 13 | \( 1 - 0.989T + 13T^{2} \) |
| 19 | \( 1 - 6.75T + 19T^{2} \) |
| 23 | \( 1 + 1.90T + 23T^{2} \) |
| 29 | \( 1 + 3.84T + 29T^{2} \) |
| 31 | \( 1 + 0.283T + 31T^{2} \) |
| 37 | \( 1 - 1.26T + 37T^{2} \) |
| 41 | \( 1 - 7.81T + 41T^{2} \) |
| 43 | \( 1 + 8.97T + 43T^{2} \) |
| 47 | \( 1 - 2.35T + 47T^{2} \) |
| 53 | \( 1 + 1.93T + 53T^{2} \) |
| 59 | \( 1 - 3.12T + 59T^{2} \) |
| 61 | \( 1 + 0.0662T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 - 8.14T + 71T^{2} \) |
| 73 | \( 1 - 1.16T + 73T^{2} \) |
| 83 | \( 1 - 4.09T + 83T^{2} \) |
| 89 | \( 1 - 16.5T + 89T^{2} \) |
| 97 | \( 1 - 6.24T + 97T^{2} \) |
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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
−8.361574789958223801296350001761, −7.77481377382815927267993953411, −7.40590201726519194625394429806, −6.42349452929768817348253044242, −5.67643618692203994188199496598, −5.07814443933830069045218800403, −3.56912945192036161878207909667, −2.32093638686005397818234625168, −1.62247014324772798672572996187, −0.75617404187452906307610613441,
0.75617404187452906307610613441, 1.62247014324772798672572996187, 2.32093638686005397818234625168, 3.56912945192036161878207909667, 5.07814443933830069045218800403, 5.67643618692203994188199496598, 6.42349452929768817348253044242, 7.40590201726519194625394429806, 7.77481377382815927267993953411, 8.361574789958223801296350001761
