| L(s) = 1 | − 3-s − 5-s + 2·7-s + 9-s − 4·13-s + 15-s − 6·17-s − 2·21-s + 25-s − 27-s − 8·29-s − 2·35-s − 4·37-s + 4·39-s + 10·41-s + 8·43-s − 45-s + 4·47-s − 3·49-s + 6·51-s + 14·53-s − 14·59-s + 6·61-s + 2·63-s + 4·65-s + 10·67-s − 6·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.10·13-s + 0.258·15-s − 1.45·17-s − 0.436·21-s + 1/5·25-s − 0.192·27-s − 1.48·29-s − 0.338·35-s − 0.657·37-s + 0.640·39-s + 1.56·41-s + 1.21·43-s − 0.149·45-s + 0.583·47-s − 3/7·49-s + 0.840·51-s + 1.92·53-s − 1.82·59-s + 0.768·61-s + 0.251·63-s + 0.496·65-s + 1.22·67-s − 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.93392707816995, −12.64710657996265, −12.26877925553386, −11.66154674016451, −11.28882097915754, −10.87894501158323, −10.67051082312468, −9.815595159333975, −9.493016592507280, −8.930969357779785, −8.481812783674223, −7.877217923060490, −7.347577273959603, −7.135417105489245, −6.572823923364987, −5.766220525347809, −5.551095643054544, −4.860155747996026, −4.392025005330646, −4.111106621495588, −3.396332955104281, −2.428621694178922, −2.259541543459024, −1.456483829632422, −0.6464534315987329, 0,
0.6464534315987329, 1.456483829632422, 2.259541543459024, 2.428621694178922, 3.396332955104281, 4.111106621495588, 4.392025005330646, 4.860155747996026, 5.551095643054544, 5.766220525347809, 6.572823923364987, 7.135417105489245, 7.347577273959603, 7.877217923060490, 8.481812783674223, 8.930969357779785, 9.493016592507280, 9.815595159333975, 10.67051082312468, 10.87894501158323, 11.28882097915754, 11.66154674016451, 12.26877925553386, 12.64710657996265, 12.93392707816995
