| L(s) = 1 | − 3-s + 5-s + 9-s + 6·13-s − 15-s + 3·17-s − 3·23-s + 25-s − 27-s + 6·29-s − 7·31-s − 8·37-s − 6·39-s − 2·41-s − 10·43-s + 45-s + 3·47-s − 7·49-s − 3·51-s + 3·53-s − 2·59-s − 7·61-s + 6·65-s − 12·67-s + 3·69-s − 12·71-s − 4·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.66·13-s − 0.258·15-s + 0.727·17-s − 0.625·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.25·31-s − 1.31·37-s − 0.960·39-s − 0.312·41-s − 1.52·43-s + 0.149·45-s + 0.437·47-s − 49-s − 0.420·51-s + 0.412·53-s − 0.260·59-s − 0.896·61-s + 0.744·65-s − 1.46·67-s + 0.361·69-s − 1.42·71-s − 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−16.22389734298478, −16.08875488049831, −15.34879260524293, −14.69430155759037, −14.07314178162324, −13.48471547697924, −13.17723056354878, −12.30371059492361, −11.95011719243195, −11.27039625870220, −10.62120464586001, −10.28560759485213, −9.622993028631531, −8.787958300545520, −8.455243792759847, −7.617680668180480, −6.905882950078665, −6.255975650221122, −5.805711624707132, −5.194737093451266, −4.406753958432179, −3.597445879325994, −3.029181802286032, −1.729243235279477, −1.302907951747630, 0,
1.302907951747630, 1.729243235279477, 3.029181802286032, 3.597445879325994, 4.406753958432179, 5.194737093451266, 5.805711624707132, 6.255975650221122, 6.905882950078665, 7.617680668180480, 8.455243792759847, 8.787958300545520, 9.622993028631531, 10.28560759485213, 10.62120464586001, 11.27039625870220, 11.95011719243195, 12.30371059492361, 13.17723056354878, 13.48471547697924, 14.07314178162324, 14.69430155759037, 15.34879260524293, 16.08875488049831, 16.22389734298478
