| L(s) = 1 | + 0.414·2-s − 1.82·4-s + 0.585·7-s − 1.58·8-s + 2.82·11-s + 0.242·14-s + 3·16-s − 4.58·17-s + 2.24·19-s + 1.17·22-s − 23-s − 1.07·28-s − 8.48·29-s − 8.48·31-s + 4.41·32-s − 1.89·34-s − 0.828·37-s + 0.928·38-s + 9.65·41-s + 10.2·43-s − 5.17·44-s − 0.414·46-s + 11.6·47-s − 6.65·49-s − 9.07·53-s − 0.928·56-s − 3.51·58-s + ⋯ |
| L(s) = 1 | + 0.292·2-s − 0.914·4-s + 0.221·7-s − 0.560·8-s + 0.852·11-s + 0.0648·14-s + 0.750·16-s − 1.11·17-s + 0.514·19-s + 0.249·22-s − 0.208·23-s − 0.202·28-s − 1.57·29-s − 1.52·31-s + 0.780·32-s − 0.325·34-s − 0.136·37-s + 0.150·38-s + 1.50·41-s + 1.56·43-s − 0.779·44-s − 0.0610·46-s + 1.70·47-s − 0.950·49-s − 1.24·53-s − 0.124·56-s − 0.461·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 0.414T + 2T^{2} \) |
| 7 | \( 1 - 0.585T + 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 4.58T + 17T^{2} \) |
| 19 | \( 1 - 2.24T + 19T^{2} \) |
| 29 | \( 1 + 8.48T + 29T^{2} \) |
| 31 | \( 1 + 8.48T + 31T^{2} \) |
| 37 | \( 1 + 0.828T + 37T^{2} \) |
| 41 | \( 1 - 9.65T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 + 9.07T + 53T^{2} \) |
| 59 | \( 1 + 3.65T + 59T^{2} \) |
| 61 | \( 1 - 4.82T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 + 2.34T + 71T^{2} \) |
| 73 | \( 1 - 9.31T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 + 1.17T + 83T^{2} \) |
| 89 | \( 1 + 1.07T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
| 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
−7.73914699908265542027964603347, −7.29387035625081921525449015230, −6.17879355020552293570222627654, −5.69411785641999205567842101980, −4.82095177899972167728205822734, −4.09171494343731411724442051418, −3.60761456529587723116679668347, −2.41719498694984989674356133806, −1.30494872758807112445736760302, 0,
1.30494872758807112445736760302, 2.41719498694984989674356133806, 3.60761456529587723116679668347, 4.09171494343731411724442051418, 4.82095177899972167728205822734, 5.69411785641999205567842101980, 6.17879355020552293570222627654, 7.29387035625081921525449015230, 7.73914699908265542027964603347
