| L(s) = 1 | − 2·5-s − 13-s − 17-s − 25-s − 2·29-s − 8·31-s − 10·37-s + 6·41-s − 4·43-s − 8·47-s − 7·49-s − 2·53-s + 4·59-s + 10·61-s + 2·65-s + 8·67-s − 12·71-s + 6·73-s + 8·79-s − 12·83-s + 2·85-s + 10·89-s + 6·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.277·13-s − 0.242·17-s − 1/5·25-s − 0.371·29-s − 1.43·31-s − 1.64·37-s + 0.937·41-s − 0.609·43-s − 1.16·47-s − 49-s − 0.274·53-s + 0.520·59-s + 1.28·61-s + 0.248·65-s + 0.977·67-s − 1.42·71-s + 0.702·73-s + 0.900·79-s − 1.31·83-s + 0.216·85-s + 1.05·89-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 127296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.79205003168546, −13.07707443510133, −12.83181331213568, −12.31303728138836, −11.75142890430988, −11.32424972239733, −11.05204191195626, −10.34982610457388, −9.890501260596545, −9.357862070243974, −8.813130510046192, −8.310000486002429, −7.883175280514848, −7.267284197958862, −6.985814215634544, −6.333907195785902, −5.677380878220368, −5.137936248263084, −4.650471069562535, −3.968232297240365, −3.516359538408118, −3.079934692771297, −2.075485627172579, −1.747947106645922, −0.6514322884366205, 0,
0.6514322884366205, 1.747947106645922, 2.075485627172579, 3.079934692771297, 3.516359538408118, 3.968232297240365, 4.650471069562535, 5.137936248263084, 5.677380878220368, 6.333907195785902, 6.985814215634544, 7.267284197958862, 7.883175280514848, 8.310000486002429, 8.813130510046192, 9.357862070243974, 9.890501260596545, 10.34982610457388, 11.05204191195626, 11.32424972239733, 11.75142890430988, 12.31303728138836, 12.83181331213568, 13.07707443510133, 13.79205003168546
