| L(s) = 1 | − 1.06·2-s − 3-s − 0.867·4-s + 2.98·5-s + 1.06·6-s − 1.35·7-s + 3.05·8-s + 9-s − 3.17·10-s + 5.67·11-s + 0.867·12-s + 3.91·13-s + 1.44·14-s − 2.98·15-s − 1.51·16-s − 17-s − 1.06·18-s − 2.63·19-s − 2.58·20-s + 1.35·21-s − 6.04·22-s − 7.94·23-s − 3.05·24-s + 3.89·25-s − 4.16·26-s − 27-s + 1.17·28-s + ⋯ |
| L(s) = 1 | − 0.752·2-s − 0.577·3-s − 0.433·4-s + 1.33·5-s + 0.434·6-s − 0.513·7-s + 1.07·8-s + 0.333·9-s − 1.00·10-s + 1.71·11-s + 0.250·12-s + 1.08·13-s + 0.386·14-s − 0.770·15-s − 0.377·16-s − 0.242·17-s − 0.250·18-s − 0.603·19-s − 0.578·20-s + 0.296·21-s − 1.28·22-s − 1.65·23-s − 0.622·24-s + 0.779·25-s − 0.817·26-s − 0.192·27-s + 0.222·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.06T + 2T^{2} \) |
| 5 | \( 1 - 2.98T + 5T^{2} \) |
| 7 | \( 1 + 1.35T + 7T^{2} \) |
| 11 | \( 1 - 5.67T + 11T^{2} \) |
| 13 | \( 1 - 3.91T + 13T^{2} \) |
| 19 | \( 1 + 2.63T + 19T^{2} \) |
| 23 | \( 1 + 7.94T + 23T^{2} \) |
| 29 | \( 1 + 10.4T + 29T^{2} \) |
| 31 | \( 1 + 1.01T + 31T^{2} \) |
| 37 | \( 1 + 6.45T + 37T^{2} \) |
| 41 | \( 1 + 1.35T + 41T^{2} \) |
| 43 | \( 1 + 5.28T + 43T^{2} \) |
| 47 | \( 1 - 0.904T + 47T^{2} \) |
| 53 | \( 1 - 6.13T + 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 - 1.90T + 61T^{2} \) |
| 67 | \( 1 - 1.30T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 - 7.18T + 73T^{2} \) |
| 83 | \( 1 + 1.91T + 83T^{2} \) |
| 89 | \( 1 - 1.08T + 89T^{2} \) |
| 97 | \( 1 + 13.0T + 97T^{2} \) |
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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
−8.362539931007051293913859476550, −7.27592510252835118061757538692, −6.40915233231759286211356167669, −6.08367483271572710879468493211, −5.29133576233425945893398456672, −4.14571842772258093041419631951, −3.65408270662097287862674505839, −1.84645667812731103609380873842, −1.47286397417260600956652291553, 0,
1.47286397417260600956652291553, 1.84645667812731103609380873842, 3.65408270662097287862674505839, 4.14571842772258093041419631951, 5.29133576233425945893398456672, 6.08367483271572710879468493211, 6.40915233231759286211356167669, 7.27592510252835118061757538692, 8.362539931007051293913859476550
