L(s) = 1 | + 2-s − 3-s + 4-s − 0.585·5-s − 6-s + 8-s + 9-s − 0.585·10-s − 3.93·11-s − 12-s − 2.37·13-s + 0.585·15-s + 16-s + 4.45·17-s + 18-s − 1.56·19-s − 0.585·20-s − 3.93·22-s + 23-s − 24-s − 4.65·25-s − 2.37·26-s − 27-s + 10.6·29-s + 0.585·30-s + 0.431·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.261·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.185·10-s − 1.18·11-s − 0.288·12-s − 0.657·13-s + 0.151·15-s + 0.250·16-s + 1.08·17-s + 0.235·18-s − 0.359·19-s − 0.130·20-s − 0.838·22-s + 0.208·23-s − 0.204·24-s − 0.931·25-s − 0.465·26-s − 0.192·27-s + 1.96·29-s + 0.106·30-s + 0.0775·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 + 0.585T + 5T^{2} \) |
| 11 | \( 1 + 3.93T + 11T^{2} \) |
| 13 | \( 1 + 2.37T + 13T^{2} \) |
| 17 | \( 1 - 4.45T + 17T^{2} \) |
| 19 | \( 1 + 1.56T + 19T^{2} \) |
| 29 | \( 1 - 10.6T + 29T^{2} \) |
| 31 | \( 1 - 0.431T + 31T^{2} \) |
| 37 | \( 1 - 4.45T + 37T^{2} \) |
| 41 | \( 1 + 9.27T + 41T^{2} \) |
| 43 | \( 1 - 5.71T + 43T^{2} \) |
| 47 | \( 1 - 7.42T + 47T^{2} \) |
| 53 | \( 1 - 7.40T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 + 15.2T + 61T^{2} \) |
| 67 | \( 1 + 4.11T + 67T^{2} \) |
| 71 | \( 1 + 11.5T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 - 3.21T + 79T^{2} \) |
| 83 | \( 1 - 7.34T + 83T^{2} \) |
| 89 | \( 1 + 16.7T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.62212144619464334759229670897, −6.86359799099450038871355670795, −5.99799656422372051308610845716, −5.52402225011793311158951394242, −4.73410228086790106923141370982, −4.24925659638848762270074605696, −3.10955300751289579886479977648, −2.55227879375180795696251839298, −1.31404839588862579550890093529, 0,
1.31404839588862579550890093529, 2.55227879375180795696251839298, 3.10955300751289579886479977648, 4.24925659638848762270074605696, 4.73410228086790106923141370982, 5.52402225011793311158951394242, 5.99799656422372051308610845716, 6.86359799099450038871355670795, 7.62212144619464334759229670897