| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s − 6·13-s − 15-s + 6·17-s + 4·19-s − 21-s + 23-s + 25-s + 27-s − 2·29-s − 4·31-s + 35-s + 2·37-s − 6·39-s + 2·41-s − 45-s + 4·47-s + 49-s + 6·51-s − 14·53-s + 4·57-s − 12·59-s + 2·61-s − 63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.66·13-s − 0.258·15-s + 1.45·17-s + 0.917·19-s − 0.218·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.718·31-s + 0.169·35-s + 0.328·37-s − 0.960·39-s + 0.312·41-s − 0.149·45-s + 0.583·47-s + 1/7·49-s + 0.840·51-s − 1.92·53-s + 0.529·57-s − 1.56·59-s + 0.256·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.98813797515399, −14.63990922480970, −14.02365227198393, −13.75963670153520, −12.80890808265788, −12.46363379934413, −12.17644185396590, −11.47238566462729, −10.88044523749930, −10.17988410199986, −9.690053243782352, −9.377098699572326, −8.784635269204509, −7.797696836108485, −7.664899006713960, −7.276175582502387, −6.454450013035041, −5.748384631645780, −5.061268688616254, −4.636028399021486, −3.729222739976865, −3.224829574039475, −2.728087923059034, −1.882285378426271, −0.9907999218270499, 0,
0.9907999218270499, 1.882285378426271, 2.728087923059034, 3.224829574039475, 3.729222739976865, 4.636028399021486, 5.061268688616254, 5.748384631645780, 6.454450013035041, 7.276175582502387, 7.664899006713960, 7.797696836108485, 8.784635269204509, 9.377098699572326, 9.690053243782352, 10.17988410199986, 10.88044523749930, 11.47238566462729, 12.17644185396590, 12.46363379934413, 12.80890808265788, 13.75963670153520, 14.02365227198393, 14.63990922480970, 14.98813797515399
