| L(s) = 1 | − 5-s + 4·7-s − 11-s − 4·13-s + 4·19-s − 6·23-s + 25-s + 6·29-s − 8·31-s − 4·35-s + 2·37-s − 6·41-s − 8·43-s + 6·47-s + 9·49-s + 6·53-s + 55-s − 12·59-s + 2·61-s + 4·65-s + 10·67-s − 12·71-s − 16·73-s − 4·77-s − 8·79-s − 6·89-s − 16·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s − 0.301·11-s − 1.10·13-s + 0.917·19-s − 1.25·23-s + 1/5·25-s + 1.11·29-s − 1.43·31-s − 0.676·35-s + 0.328·37-s − 0.937·41-s − 1.21·43-s + 0.875·47-s + 9/7·49-s + 0.824·53-s + 0.134·55-s − 1.56·59-s + 0.256·61-s + 0.496·65-s + 1.22·67-s − 1.42·71-s − 1.87·73-s − 0.455·77-s − 0.900·79-s − 0.635·89-s − 1.67·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−7.42958001814686298772216858390, −7.17885705414315583501683816518, −5.97766150450361195120091666391, −5.24043083671655618876266346074, −4.74399308506065368743195948188, −4.06129546619223126731419837922, −3.06771808978900449398252647170, −2.15971418307743606157364508126, −1.35422897932614332939614819531, 0,
1.35422897932614332939614819531, 2.15971418307743606157364508126, 3.06771808978900449398252647170, 4.06129546619223126731419837922, 4.74399308506065368743195948188, 5.24043083671655618876266346074, 5.97766150450361195120091666391, 7.17885705414315583501683816518, 7.42958001814686298772216858390
