| L(s) = 1 | + 2·5-s + 32·7-s − 68·11-s + 13·13-s + 14·17-s − 4·19-s + 72·23-s − 121·25-s − 102·29-s + 136·31-s + 64·35-s − 386·37-s − 250·41-s + 140·43-s − 296·47-s + 681·49-s − 526·53-s − 136·55-s + 332·59-s − 410·61-s + 26·65-s − 596·67-s − 880·71-s + 506·73-s − 2.17e3·77-s + 640·79-s + 1.38e3·83-s + ⋯ |
| L(s) = 1 | + 0.178·5-s + 1.72·7-s − 1.86·11-s + 0.277·13-s + 0.199·17-s − 0.0482·19-s + 0.652·23-s − 0.967·25-s − 0.653·29-s + 0.787·31-s + 0.309·35-s − 1.71·37-s − 0.952·41-s + 0.496·43-s − 0.918·47-s + 1.98·49-s − 1.36·53-s − 0.333·55-s + 0.732·59-s − 0.860·61-s + 0.0496·65-s − 1.08·67-s − 1.47·71-s + 0.811·73-s − 3.22·77-s + 0.911·79-s + 1.82·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - p T \) |
| good | 5 | \( 1 - 2 T + p^{3} T^{2} \) |
| 7 | \( 1 - 32 T + p^{3} T^{2} \) |
| 11 | \( 1 + 68 T + p^{3} T^{2} \) |
| 17 | \( 1 - 14 T + p^{3} T^{2} \) |
| 19 | \( 1 + 4 T + p^{3} T^{2} \) |
| 23 | \( 1 - 72 T + p^{3} T^{2} \) |
| 29 | \( 1 + 102 T + p^{3} T^{2} \) |
| 31 | \( 1 - 136 T + p^{3} T^{2} \) |
| 37 | \( 1 + 386 T + p^{3} T^{2} \) |
| 41 | \( 1 + 250 T + p^{3} T^{2} \) |
| 43 | \( 1 - 140 T + p^{3} T^{2} \) |
| 47 | \( 1 + 296 T + p^{3} T^{2} \) |
| 53 | \( 1 + 526 T + p^{3} T^{2} \) |
| 59 | \( 1 - 332 T + p^{3} T^{2} \) |
| 61 | \( 1 + 410 T + p^{3} T^{2} \) |
| 67 | \( 1 + 596 T + p^{3} T^{2} \) |
| 71 | \( 1 + 880 T + p^{3} T^{2} \) |
| 73 | \( 1 - 506 T + p^{3} T^{2} \) |
| 79 | \( 1 - 640 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1380 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1450 T + p^{3} T^{2} \) |
| 97 | \( 1 + 446 T + p^{3} T^{2} \) |
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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
−8.195388486208075998154937620857, −7.937445137536748659362940740000, −7.10331963659854396635852329695, −5.84354393942905810020045919883, −5.13632314981167661029395767773, −4.65329337091138872322091420891, −3.34647263876483323827317530601, −2.24560772142921860259271377891, −1.45108895516746132322810241216, 0,
1.45108895516746132322810241216, 2.24560772142921860259271377891, 3.34647263876483323827317530601, 4.65329337091138872322091420891, 5.13632314981167661029395767773, 5.84354393942905810020045919883, 7.10331963659854396635852329695, 7.937445137536748659362940740000, 8.195388486208075998154937620857
