| L(s) = 1 | − 3-s + 5-s + 9-s − 4·11-s − 13-s − 15-s + 17-s + 8·19-s − 8·23-s + 25-s − 27-s − 6·29-s + 4·33-s + 6·37-s + 39-s + 6·41-s − 4·43-s + 45-s + 4·47-s − 7·49-s − 51-s + 10·53-s − 4·55-s − 8·57-s − 6·61-s − 65-s + 8·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.277·13-s − 0.258·15-s + 0.242·17-s + 1.83·19-s − 1.66·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.696·33-s + 0.986·37-s + 0.160·39-s + 0.937·41-s − 0.609·43-s + 0.149·45-s + 0.583·47-s − 49-s − 0.140·51-s + 1.37·53-s − 0.539·55-s − 1.05·57-s − 0.768·61-s − 0.124·65-s + 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.19708382752526, −12.82852078190431, −12.27445445474322, −11.68464248322276, −11.59953681984438, −10.80036898292212, −10.45887121239075, −9.958273582833800, −9.579383394495299, −9.235989254149079, −8.425541906764383, −7.846025009676550, −7.533414152461404, −7.206990876646244, −6.316265019737871, −5.940537006863478, −5.528448231453618, −5.082677028333251, −4.610132461158393, −3.844957345470398, −3.369214454807361, −2.608376617000022, −2.191746899090918, −1.450978609786027, −0.7416042806773189, 0,
0.7416042806773189, 1.450978609786027, 2.191746899090918, 2.608376617000022, 3.369214454807361, 3.844957345470398, 4.610132461158393, 5.082677028333251, 5.528448231453618, 5.940537006863478, 6.316265019737871, 7.206990876646244, 7.533414152461404, 7.846025009676550, 8.425541906764383, 9.235989254149079, 9.579383394495299, 9.958273582833800, 10.45887121239075, 10.80036898292212, 11.59953681984438, 11.68464248322276, 12.27445445474322, 12.82852078190431, 13.19708382752526
