| L(s) = 1 | + 0.648·3-s + 2.57·5-s + 1.91·7-s − 2.57·9-s + 2.67·11-s − 2.76·13-s + 1.67·15-s − 5.56·17-s − 5.65·19-s + 1.24·21-s + 1.28·23-s + 1.64·25-s − 3.61·27-s − 3.23·29-s − 2.44·31-s + 1.73·33-s + 4.93·35-s − 1.30·37-s − 1.78·39-s + 8.94·41-s − 2.24·43-s − 6.65·45-s − 10.6·47-s − 3.33·49-s − 3.60·51-s + 7.69·53-s + 6.89·55-s + ⋯ |
| L(s) = 1 | + 0.374·3-s + 1.15·5-s + 0.723·7-s − 0.859·9-s + 0.805·11-s − 0.765·13-s + 0.431·15-s − 1.35·17-s − 1.29·19-s + 0.270·21-s + 0.267·23-s + 0.329·25-s − 0.696·27-s − 0.600·29-s − 0.439·31-s + 0.301·33-s + 0.834·35-s − 0.215·37-s − 0.286·39-s + 1.39·41-s − 0.342·43-s − 0.991·45-s − 1.55·47-s − 0.476·49-s − 0.505·51-s + 1.05·53-s + 0.929·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - 0.648T + 3T^{2} \) |
| 5 | \( 1 - 2.57T + 5T^{2} \) |
| 7 | \( 1 - 1.91T + 7T^{2} \) |
| 11 | \( 1 - 2.67T + 11T^{2} \) |
| 13 | \( 1 + 2.76T + 13T^{2} \) |
| 17 | \( 1 + 5.56T + 17T^{2} \) |
| 19 | \( 1 + 5.65T + 19T^{2} \) |
| 23 | \( 1 - 1.28T + 23T^{2} \) |
| 29 | \( 1 + 3.23T + 29T^{2} \) |
| 31 | \( 1 + 2.44T + 31T^{2} \) |
| 37 | \( 1 + 1.30T + 37T^{2} \) |
| 41 | \( 1 - 8.94T + 41T^{2} \) |
| 43 | \( 1 + 2.24T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 - 7.69T + 53T^{2} \) |
| 59 | \( 1 + 4.71T + 59T^{2} \) |
| 61 | \( 1 - 8.59T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 + 9.07T + 71T^{2} \) |
| 73 | \( 1 + 2.09T + 73T^{2} \) |
| 79 | \( 1 - 2.24T + 79T^{2} \) |
| 83 | \( 1 + 16.3T + 83T^{2} \) |
| 89 | \( 1 - 3.73T + 89T^{2} \) |
| 97 | \( 1 + 1.84T + 97T^{2} \) |
| 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
−7.49826391755804260074367752639, −6.68295029991214673417453392547, −6.12424914503951735594560019159, −5.45156970843110163197088155501, −4.66560823341102374125254140177, −4.00545385804032654184789745629, −2.85091584199236084070510874480, −2.14860367026451110646051282708, −1.62868544173476232198497670342, 0,
1.62868544173476232198497670342, 2.14860367026451110646051282708, 2.85091584199236084070510874480, 4.00545385804032654184789745629, 4.66560823341102374125254140177, 5.45156970843110163197088155501, 6.12424914503951735594560019159, 6.68295029991214673417453392547, 7.49826391755804260074367752639
