| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s − 3·11-s − 3·13-s − 15-s + 17-s − 3·19-s − 21-s + 7·23-s − 4·25-s + 27-s + 6·29-s + 10·31-s − 3·33-s + 35-s − 4·37-s − 3·39-s − 9·41-s − 9·43-s − 45-s + 6·47-s + 49-s + 51-s + 10·53-s + 3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.904·11-s − 0.832·13-s − 0.258·15-s + 0.242·17-s − 0.688·19-s − 0.218·21-s + 1.45·23-s − 4/5·25-s + 0.192·27-s + 1.11·29-s + 1.79·31-s − 0.522·33-s + 0.169·35-s − 0.657·37-s − 0.480·39-s − 1.40·41-s − 1.37·43-s − 0.149·45-s + 0.875·47-s + 1/7·49-s + 0.140·51-s + 1.37·53-s + 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−15.50630589277525, −15.29467805689539, −14.89235848110905, −13.97722750511358, −13.69211885831164, −13.05810079428831, −12.58641112573371, −11.94346943361960, −11.57811766499747, −10.65178786886937, −10.08269695112832, −9.961774321800147, −8.969923711197601, −8.455058638360948, −8.073158521153884, −7.308642367417982, −6.873182738699420, −6.246706025019926, −5.212170384384418, −4.887201099453239, −4.093801269317643, −3.331227341721045, −2.743088067088669, −2.177952361718039, −1.008018720802396, 0,
1.008018720802396, 2.177952361718039, 2.743088067088669, 3.331227341721045, 4.093801269317643, 4.887201099453239, 5.212170384384418, 6.246706025019926, 6.873182738699420, 7.308642367417982, 8.073158521153884, 8.455058638360948, 8.969923711197601, 9.961774321800147, 10.08269695112832, 10.65178786886937, 11.57811766499747, 11.94346943361960, 12.58641112573371, 13.05810079428831, 13.69211885831164, 13.97722750511358, 14.89235848110905, 15.29467805689539, 15.50630589277525
