| L(s) = 1 | − 0.308·2-s − 1.39·3-s − 1.90·4-s + 2.83·5-s + 0.431·6-s + 4.16·7-s + 1.20·8-s − 1.04·9-s − 0.874·10-s + 1.93·11-s + 2.66·12-s + 4.12·13-s − 1.28·14-s − 3.96·15-s + 3.43·16-s + 1.03·17-s + 0.322·18-s − 7.34·19-s − 5.39·20-s − 5.82·21-s − 0.596·22-s − 4.09·23-s − 1.68·24-s + 3.02·25-s − 1.27·26-s + 5.65·27-s − 7.93·28-s + ⋯ |
| L(s) = 1 | − 0.218·2-s − 0.807·3-s − 0.952·4-s + 1.26·5-s + 0.176·6-s + 1.57·7-s + 0.426·8-s − 0.348·9-s − 0.276·10-s + 0.582·11-s + 0.768·12-s + 1.14·13-s − 0.343·14-s − 1.02·15-s + 0.859·16-s + 0.250·17-s + 0.0760·18-s − 1.68·19-s − 1.20·20-s − 1.27·21-s − 0.127·22-s − 0.854·23-s − 0.344·24-s + 0.604·25-s − 0.250·26-s + 1.08·27-s − 1.50·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8829406305\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8829406305\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 139 | \( 1 - T \) |
| good | 2 | \( 1 + 0.308T + 2T^{2} \) |
| 3 | \( 1 + 1.39T + 3T^{2} \) |
| 5 | \( 1 - 2.83T + 5T^{2} \) |
| 7 | \( 1 - 4.16T + 7T^{2} \) |
| 11 | \( 1 - 1.93T + 11T^{2} \) |
| 13 | \( 1 - 4.12T + 13T^{2} \) |
| 17 | \( 1 - 1.03T + 17T^{2} \) |
| 19 | \( 1 + 7.34T + 19T^{2} \) |
| 23 | \( 1 + 4.09T + 23T^{2} \) |
| 29 | \( 1 - 4.49T + 29T^{2} \) |
| 31 | \( 1 + 1.28T + 31T^{2} \) |
| 37 | \( 1 + 7.77T + 37T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 43 | \( 1 + 7.87T + 43T^{2} \) |
| 47 | \( 1 + 6.84T + 47T^{2} \) |
| 53 | \( 1 - 3.68T + 53T^{2} \) |
| 59 | \( 1 - 1.24T + 59T^{2} \) |
| 61 | \( 1 - 0.281T + 61T^{2} \) |
| 67 | \( 1 + 8.51T + 67T^{2} \) |
| 71 | \( 1 + 6.40T + 71T^{2} \) |
| 73 | \( 1 + 5.92T + 73T^{2} \) |
| 79 | \( 1 + 8.65T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 + 3.23T + 89T^{2} \) |
| 97 | \( 1 + 7.27T + 97T^{2} \) |
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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
−13.32731619225253505087244368274, −12.08233218630803587058427298945, −10.97436313414195187164114782786, −10.25743019313060138157237943331, −8.874193127198479613628287545857, −8.264283930550231462388777861860, −6.28236602205375346718529176661, −5.44580997586798804423401857281, −4.34532943215737786972727556586, −1.58075068472232777858525680636,
1.58075068472232777858525680636, 4.34532943215737786972727556586, 5.44580997586798804423401857281, 6.28236602205375346718529176661, 8.264283930550231462388777861860, 8.874193127198479613628287545857, 10.25743019313060138157237943331, 10.97436313414195187164114782786, 12.08233218630803587058427298945, 13.32731619225253505087244368274
