| L(s) = 1 | + 2-s + 0.689·3-s + 4-s − 0.213·5-s + 0.689·6-s − 0.113·7-s + 8-s − 2.52·9-s − 0.213·10-s + 2.79·11-s + 0.689·12-s − 4.76·13-s − 0.113·14-s − 0.146·15-s + 16-s + 0.198·17-s − 2.52·18-s − 4.53·19-s − 0.213·20-s − 0.0779·21-s + 2.79·22-s − 5.87·23-s + 0.689·24-s − 4.95·25-s − 4.76·26-s − 3.80·27-s − 0.113·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.398·3-s + 0.5·4-s − 0.0952·5-s + 0.281·6-s − 0.0427·7-s + 0.353·8-s − 0.841·9-s − 0.0673·10-s + 0.843·11-s + 0.199·12-s − 1.32·13-s − 0.0302·14-s − 0.0379·15-s + 0.250·16-s + 0.0481·17-s − 0.595·18-s − 1.03·19-s − 0.0476·20-s − 0.0170·21-s + 0.596·22-s − 1.22·23-s + 0.140·24-s − 0.990·25-s − 0.934·26-s − 0.733·27-s − 0.0213·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 2011 | \( 1 - T \) |
| good | 3 | \( 1 - 0.689T + 3T^{2} \) |
| 5 | \( 1 + 0.213T + 5T^{2} \) |
| 7 | \( 1 + 0.113T + 7T^{2} \) |
| 11 | \( 1 - 2.79T + 11T^{2} \) |
| 13 | \( 1 + 4.76T + 13T^{2} \) |
| 17 | \( 1 - 0.198T + 17T^{2} \) |
| 19 | \( 1 + 4.53T + 19T^{2} \) |
| 23 | \( 1 + 5.87T + 23T^{2} \) |
| 29 | \( 1 - 4.91T + 29T^{2} \) |
| 31 | \( 1 - 3.25e - 5T + 31T^{2} \) |
| 37 | \( 1 - 7.31T + 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 - 1.63T + 43T^{2} \) |
| 47 | \( 1 + 0.417T + 47T^{2} \) |
| 53 | \( 1 + 5.99T + 53T^{2} \) |
| 59 | \( 1 - 1.97T + 59T^{2} \) |
| 61 | \( 1 + 2.28T + 61T^{2} \) |
| 67 | \( 1 + 3.43T + 67T^{2} \) |
| 71 | \( 1 + 3.22T + 71T^{2} \) |
| 73 | \( 1 - 4.81T + 73T^{2} \) |
| 79 | \( 1 + 6.40T + 79T^{2} \) |
| 83 | \( 1 + 1.84T + 83T^{2} \) |
| 89 | \( 1 + 9.05T + 89T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
| 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
−8.081663330277093486716921744161, −7.33367290822970678145008949291, −6.40290690846486429340193291862, −5.96918852401231292367057422088, −4.93205245768911747047164517384, −4.26314452442575738191063389325, −3.44519417168935714676298635487, −2.56946035652360188388219154880, −1.81346876547379779344054640636, 0,
1.81346876547379779344054640636, 2.56946035652360188388219154880, 3.44519417168935714676298635487, 4.26314452442575738191063389325, 4.93205245768911747047164517384, 5.96918852401231292367057422088, 6.40290690846486429340193291862, 7.33367290822970678145008949291, 8.081663330277093486716921744161
