| L(s) = 1 | − 3-s − 2·5-s + 9-s + 6·11-s − 4·13-s + 2·15-s − 17-s − 6·19-s − 4·23-s − 25-s − 27-s + 8·29-s − 6·33-s + 4·39-s + 2·41-s + 4·43-s − 2·45-s + 51-s − 2·53-s − 12·55-s + 6·57-s − 6·59-s + 2·61-s + 8·65-s + 8·67-s + 4·69-s − 10·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s + 1.80·11-s − 1.10·13-s + 0.516·15-s − 0.242·17-s − 1.37·19-s − 0.834·23-s − 1/5·25-s − 0.192·27-s + 1.48·29-s − 1.04·33-s + 0.640·39-s + 0.312·41-s + 0.609·43-s − 0.298·45-s + 0.140·51-s − 0.274·53-s − 1.61·55-s + 0.794·57-s − 0.781·59-s + 0.256·61-s + 0.992·65-s + 0.977·67-s + 0.481·69-s − 1.17·73-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 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 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 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.55931672073390, −12.68574496424650, −12.53875314332237, −12.01867992640523, −11.62854258152671, −11.39443336927293, −10.64849012437906, −10.27427187173248, −9.709443171884058, −9.217815388086073, −8.685913980283053, −8.218171796351262, −7.633806245427260, −7.168201086884039, −6.518811549397785, −6.374665729600998, −5.730096884758586, −4.928998001340917, −4.354119339450943, −4.192985424825633, −3.627051273191039, −2.807267127510319, −2.128849835610753, −1.477335708741470, −0.6753275361545359, 0,
0.6753275361545359, 1.477335708741470, 2.128849835610753, 2.807267127510319, 3.627051273191039, 4.192985424825633, 4.354119339450943, 4.928998001340917, 5.730096884758586, 6.374665729600998, 6.518811549397785, 7.168201086884039, 7.633806245427260, 8.218171796351262, 8.685913980283053, 9.217815388086073, 9.709443171884058, 10.27427187173248, 10.64849012437906, 11.39443336927293, 11.62854258152671, 12.01867992640523, 12.53875314332237, 12.68574496424650, 13.55931672073390
