| L(s) = 1 | + 2.59·2-s + 3-s + 4.75·4-s − 1.90·5-s + 2.59·6-s + 1.92·7-s + 7.17·8-s + 9-s − 4.94·10-s − 0.601·11-s + 4.75·12-s + 3.49·13-s + 5.00·14-s − 1.90·15-s + 9.12·16-s + 17-s + 2.59·18-s − 2.11·19-s − 9.04·20-s + 1.92·21-s − 1.56·22-s + 6.97·23-s + 7.17·24-s − 1.38·25-s + 9.07·26-s + 27-s + 9.15·28-s + ⋯ |
| L(s) = 1 | + 1.83·2-s + 0.577·3-s + 2.37·4-s − 0.850·5-s + 1.06·6-s + 0.727·7-s + 2.53·8-s + 0.333·9-s − 1.56·10-s − 0.181·11-s + 1.37·12-s + 0.968·13-s + 1.33·14-s − 0.490·15-s + 2.28·16-s + 0.242·17-s + 0.612·18-s − 0.485·19-s − 2.02·20-s + 0.419·21-s − 0.333·22-s + 1.45·23-s + 1.46·24-s − 0.276·25-s + 1.78·26-s + 0.192·27-s + 1.72·28-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
| \(L(1)\) |
\(\approx\) |
\(7.718778418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.718778418\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.59T + 2T^{2} \) |
| 5 | \( 1 + 1.90T + 5T^{2} \) |
| 7 | \( 1 - 1.92T + 7T^{2} \) |
| 11 | \( 1 + 0.601T + 11T^{2} \) |
| 13 | \( 1 - 3.49T + 13T^{2} \) |
| 19 | \( 1 + 2.11T + 19T^{2} \) |
| 23 | \( 1 - 6.97T + 23T^{2} \) |
| 29 | \( 1 + 0.266T + 29T^{2} \) |
| 31 | \( 1 - 4.68T + 31T^{2} \) |
| 37 | \( 1 + 6.70T + 37T^{2} \) |
| 41 | \( 1 - 2.69T + 41T^{2} \) |
| 43 | \( 1 + 2.92T + 43T^{2} \) |
| 47 | \( 1 + 7.17T + 47T^{2} \) |
| 53 | \( 1 - 9.70T + 53T^{2} \) |
| 59 | \( 1 - 0.899T + 59T^{2} \) |
| 61 | \( 1 - 5.08T + 61T^{2} \) |
| 67 | \( 1 + 1.11T + 67T^{2} \) |
| 71 | \( 1 - 2.37T + 71T^{2} \) |
| 73 | \( 1 + 4.92T + 73T^{2} \) |
| 83 | \( 1 + 8.55T + 83T^{2} \) |
| 89 | \( 1 - 9.51T + 89T^{2} \) |
| 97 | \( 1 - 0.679T + 97T^{2} \) |
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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
−8.250571752776102876290493731511, −7.52485177397647149983848150477, −6.87335713402766514592657608769, −6.10547688103068354976059711146, −5.15619410310318740489071663151, −4.62740097035166910109340732873, −3.80474565619337678888881556248, −3.32949706048997908482899943629, −2.39163576316242421477086824206, −1.34716091482095992893725062918,
1.34716091482095992893725062918, 2.39163576316242421477086824206, 3.32949706048997908482899943629, 3.80474565619337678888881556248, 4.62740097035166910109340732873, 5.15619410310318740489071663151, 6.10547688103068354976059711146, 6.87335713402766514592657608769, 7.52485177397647149983848150477, 8.250571752776102876290493731511
