| L(s) = 1 | + 2-s − 3-s + 4-s + 2.24·5-s − 6-s − 7-s + 8-s + 9-s + 2.24·10-s − 0.692·11-s − 12-s − 14-s − 2.24·15-s + 16-s − 2.44·17-s + 18-s − 3.40·19-s + 2.24·20-s + 21-s − 0.692·22-s − 0.753·23-s − 24-s + 0.0489·25-s − 27-s − 28-s − 3.66·29-s − 2.24·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.00·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.710·10-s − 0.208·11-s − 0.288·12-s − 0.267·14-s − 0.580·15-s + 0.250·16-s − 0.593·17-s + 0.235·18-s − 0.781·19-s + 0.502·20-s + 0.218·21-s − 0.147·22-s − 0.157·23-s − 0.204·24-s + 0.00978·25-s − 0.192·27-s − 0.188·28-s − 0.680·29-s − 0.410·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 2.24T + 5T^{2} \) |
| 11 | \( 1 + 0.692T + 11T^{2} \) |
| 17 | \( 1 + 2.44T + 17T^{2} \) |
| 19 | \( 1 + 3.40T + 19T^{2} \) |
| 23 | \( 1 + 0.753T + 23T^{2} \) |
| 29 | \( 1 + 3.66T + 29T^{2} \) |
| 31 | \( 1 - 4.40T + 31T^{2} \) |
| 37 | \( 1 + 6.96T + 37T^{2} \) |
| 41 | \( 1 + 5.63T + 41T^{2} \) |
| 43 | \( 1 + 1.64T + 43T^{2} \) |
| 47 | \( 1 + 2.10T + 47T^{2} \) |
| 53 | \( 1 + 1.69T + 53T^{2} \) |
| 59 | \( 1 + 4.96T + 59T^{2} \) |
| 61 | \( 1 + 5.09T + 61T^{2} \) |
| 67 | \( 1 + 7.84T + 67T^{2} \) |
| 71 | \( 1 - 8.76T + 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 79 | \( 1 - 1.97T + 79T^{2} \) |
| 83 | \( 1 - 9.22T + 83T^{2} \) |
| 89 | \( 1 - 3.44T + 89T^{2} \) |
| 97 | \( 1 + 4.46T + 97T^{2} \) |
| 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
−7.32712859037496336430154832223, −6.51716860082565932167643268317, −6.23820194652244149235027527758, −5.47085664785600040572254088755, −4.87675568874690410867041446476, −4.09247443070998543862701885464, −3.18717598740439187574770905744, −2.24076148305167287093340309090, −1.55825808918224414579285535785, 0,
1.55825808918224414579285535785, 2.24076148305167287093340309090, 3.18717598740439187574770905744, 4.09247443070998543862701885464, 4.87675568874690410867041446476, 5.47085664785600040572254088755, 6.23820194652244149235027527758, 6.51716860082565932167643268317, 7.32712859037496336430154832223
