| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 2.96·7-s − 8-s + 9-s − 3.35·11-s + 12-s − 4.96·13-s − 2.96·14-s + 16-s + 1.35·17-s − 18-s − 4.96·19-s + 2.96·21-s + 3.35·22-s − 23-s − 24-s + 4.96·26-s + 27-s + 2.96·28-s + 7.73·29-s − 4·31-s − 32-s − 3.35·33-s − 1.35·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s + 1.11·7-s − 0.353·8-s + 0.333·9-s − 1.01·11-s + 0.288·12-s − 1.37·13-s − 0.791·14-s + 0.250·16-s + 0.327·17-s − 0.235·18-s − 1.13·19-s + 0.646·21-s + 0.714·22-s − 0.208·23-s − 0.204·24-s + 0.973·26-s + 0.192·27-s + 0.559·28-s + 1.43·29-s − 0.718·31-s − 0.176·32-s − 0.583·33-s − 0.231·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 - 2.96T + 7T^{2} \) |
| 11 | \( 1 + 3.35T + 11T^{2} \) |
| 13 | \( 1 + 4.96T + 13T^{2} \) |
| 17 | \( 1 - 1.35T + 17T^{2} \) |
| 19 | \( 1 + 4.96T + 19T^{2} \) |
| 29 | \( 1 - 7.73T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 7.61T + 37T^{2} \) |
| 41 | \( 1 - 4.70T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 - 3.22T + 47T^{2} \) |
| 53 | \( 1 + 6.96T + 53T^{2} \) |
| 59 | \( 1 - 1.22T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 + 7.61T + 67T^{2} \) |
| 71 | \( 1 + 2.18T + 71T^{2} \) |
| 73 | \( 1 + 9.92T + 73T^{2} \) |
| 79 | \( 1 + 4.12T + 79T^{2} \) |
| 83 | \( 1 + 6.38T + 83T^{2} \) |
| 89 | \( 1 - 9.92T + 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
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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
−8.283013254583847062558932925473, −7.62009069527465232948193517951, −7.13190996567842549811751982294, −6.06324685912821118957369484650, −4.99962534891339095194888744620, −4.54692251220821180241167425042, −3.17972308835723140094762879940, −2.34525011462089560304973847153, −1.61446291174717628369372889735, 0,
1.61446291174717628369372889735, 2.34525011462089560304973847153, 3.17972308835723140094762879940, 4.54692251220821180241167425042, 4.99962534891339095194888744620, 6.06324685912821118957369484650, 7.13190996567842549811751982294, 7.62009069527465232948193517951, 8.283013254583847062558932925473
