| L(s) = 1 | − 8·3-s + 5·5-s + 4·7-s + 37·9-s − 20·11-s − 10·13-s − 40·15-s − 62·17-s + 8·19-s − 32·21-s + 192·23-s + 25·25-s − 80·27-s − 154·29-s + 124·31-s + 160·33-s + 20·35-s + 37·37-s + 80·39-s + 186·41-s + 92·43-s + 185·45-s + 476·47-s − 327·49-s + 496·51-s − 258·53-s − 100·55-s + ⋯ |
| L(s) = 1 | − 1.53·3-s + 0.447·5-s + 0.215·7-s + 1.37·9-s − 0.548·11-s − 0.213·13-s − 0.688·15-s − 0.884·17-s + 0.0965·19-s − 0.332·21-s + 1.74·23-s + 1/5·25-s − 0.570·27-s − 0.986·29-s + 0.718·31-s + 0.844·33-s + 0.0965·35-s + 0.164·37-s + 0.328·39-s + 0.708·41-s + 0.326·43-s + 0.612·45-s + 1.47·47-s − 0.953·49-s + 1.36·51-s − 0.668·53-s − 0.245·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 37 | \( 1 - p T \) |
| good | 3 | \( 1 + 8 T + p^{3} T^{2} \) |
| 7 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 + 20 T + p^{3} T^{2} \) |
| 13 | \( 1 + 10 T + p^{3} T^{2} \) |
| 17 | \( 1 + 62 T + p^{3} T^{2} \) |
| 19 | \( 1 - 8 T + p^{3} T^{2} \) |
| 23 | \( 1 - 192 T + p^{3} T^{2} \) |
| 29 | \( 1 + 154 T + p^{3} T^{2} \) |
| 31 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 41 | \( 1 - 186 T + p^{3} T^{2} \) |
| 43 | \( 1 - 92 T + p^{3} T^{2} \) |
| 47 | \( 1 - 476 T + p^{3} T^{2} \) |
| 53 | \( 1 + 258 T + p^{3} T^{2} \) |
| 59 | \( 1 + 176 T + p^{3} T^{2} \) |
| 61 | \( 1 + 458 T + p^{3} T^{2} \) |
| 67 | \( 1 - 336 T + p^{3} T^{2} \) |
| 71 | \( 1 + 232 T + p^{3} T^{2} \) |
| 73 | \( 1 + 470 T + p^{3} T^{2} \) |
| 79 | \( 1 + 676 T + p^{3} T^{2} \) |
| 83 | \( 1 + 608 T + p^{3} T^{2} \) |
| 89 | \( 1 + 102 T + p^{3} T^{2} \) |
| 97 | \( 1 + 30 T + p^{3} T^{2} \) |
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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
−9.709980029515795583295850984572, −8.849122049786096672365681603392, −7.54820859351736399026364923981, −6.74360585180789760453569142101, −5.88785569470062002210102315622, −5.14530903165549573481897656362, −4.40319655000327813824121463414, −2.70430894877784115189691868195, −1.24665132713857939188661949730, 0,
1.24665132713857939188661949730, 2.70430894877784115189691868195, 4.40319655000327813824121463414, 5.14530903165549573481897656362, 5.88785569470062002210102315622, 6.74360585180789760453569142101, 7.54820859351736399026364923981, 8.849122049786096672365681603392, 9.709980029515795583295850984572
