| L(s) = 1 | − 3-s − 2·5-s + 9-s − 4·11-s + 6·13-s + 2·15-s − 17-s − 25-s − 27-s − 2·29-s + 4·33-s − 6·37-s − 6·39-s − 2·41-s + 4·43-s − 2·45-s + 51-s + 2·53-s + 8·55-s + 4·59-s + 2·61-s − 12·65-s + 4·67-s − 14·73-s + 75-s − 8·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s − 1.20·11-s + 1.66·13-s + 0.516·15-s − 0.242·17-s − 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.696·33-s − 0.986·37-s − 0.960·39-s − 0.312·41-s + 0.609·43-s − 0.298·45-s + 0.140·51-s + 0.274·53-s + 1.07·55-s + 0.520·59-s + 0.256·61-s − 1.48·65-s + 0.488·67-s − 1.63·73-s + 0.115·75-s − 0.900·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−13.36524619886642, −12.99134990278379, −12.66493846135404, −11.93868552553020, −11.59697940174923, −11.13931628203967, −10.79892864517201, −10.27675854236545, −9.900037369742050, −9.054352645250855, −8.612631955503100, −8.227362178122030, −7.703201800733051, −7.182145770875517, −6.770633485670534, −5.967120626375592, −5.728210149713069, −5.151767130167265, −4.462110113796702, −4.037353684797965, −3.473750929983063, −2.965574418264827, −2.118283157645020, −1.457445925274305, −0.6584608898166502, 0,
0.6584608898166502, 1.457445925274305, 2.118283157645020, 2.965574418264827, 3.473750929983063, 4.037353684797965, 4.462110113796702, 5.151767130167265, 5.728210149713069, 5.967120626375592, 6.770633485670534, 7.182145770875517, 7.703201800733051, 8.227362178122030, 8.612631955503100, 9.054352645250855, 9.900037369742050, 10.27675854236545, 10.79892864517201, 11.13931628203967, 11.59697940174923, 11.93868552553020, 12.66493846135404, 12.99134990278379, 13.36524619886642
