| L(s) = 1 | − 2.38·2-s + 3-s + 3.67·4-s + 4.36·5-s − 2.38·6-s − 4.18·7-s − 3.98·8-s + 9-s − 10.4·10-s + 1.39·11-s + 3.67·12-s − 0.773·13-s + 9.96·14-s + 4.36·15-s + 2.14·16-s − 17-s − 2.38·18-s − 7.43·19-s + 16.0·20-s − 4.18·21-s − 3.32·22-s − 3.06·23-s − 3.98·24-s + 14.0·25-s + 1.84·26-s + 27-s − 15.3·28-s + ⋯ |
| L(s) = 1 | − 1.68·2-s + 0.577·3-s + 1.83·4-s + 1.95·5-s − 0.972·6-s − 1.58·7-s − 1.40·8-s + 0.333·9-s − 3.29·10-s + 0.420·11-s + 1.06·12-s − 0.214·13-s + 2.66·14-s + 1.12·15-s + 0.535·16-s − 0.242·17-s − 0.561·18-s − 1.70·19-s + 3.58·20-s − 0.913·21-s − 0.708·22-s − 0.638·23-s − 0.813·24-s + 2.81·25-s + 0.361·26-s + 0.192·27-s − 2.90·28-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 | \( 1 + 2.38T + 2T^{2} \) |
| 5 | \( 1 - 4.36T + 5T^{2} \) |
| 7 | \( 1 + 4.18T + 7T^{2} \) |
| 11 | \( 1 - 1.39T + 11T^{2} \) |
| 13 | \( 1 + 0.773T + 13T^{2} \) |
| 19 | \( 1 + 7.43T + 19T^{2} \) |
| 23 | \( 1 + 3.06T + 23T^{2} \) |
| 29 | \( 1 - 1.78T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 + 3.14T + 37T^{2} \) |
| 41 | \( 1 + 2.80T + 41T^{2} \) |
| 43 | \( 1 - 5.21T + 43T^{2} \) |
| 47 | \( 1 - 0.527T + 47T^{2} \) |
| 53 | \( 1 - 2.73T + 53T^{2} \) |
| 59 | \( 1 - 4.55T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 - 9.39T + 67T^{2} \) |
| 71 | \( 1 + 1.25T + 71T^{2} \) |
| 73 | \( 1 + 9.12T + 73T^{2} \) |
| 83 | \( 1 + 0.320T + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 - 7.35T + 97T^{2} \) |
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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
−8.515471372301511818876739300143, −7.35016485303923955937165345611, −6.59293597423898782235816170196, −6.37523992769933270681887716303, −5.48643196238787894015967364366, −4.02380702330435935024996524017, −2.80891647394359007680709757197, −2.21404981580866783698986055771, −1.48898493518494731399006930076, 0,
1.48898493518494731399006930076, 2.21404981580866783698986055771, 2.80891647394359007680709757197, 4.02380702330435935024996524017, 5.48643196238787894015967364366, 6.37523992769933270681887716303, 6.59293597423898782235816170196, 7.35016485303923955937165345611, 8.515471372301511818876739300143
