| L(s) = 1 | + 2.14·2-s + 2.60·4-s − 2.74·7-s + 1.29·8-s + 3.74·11-s − 6.29·13-s − 5.89·14-s − 2.43·16-s + 19-s + 8.03·22-s + 0.543·23-s − 13.4·26-s − 7.14·28-s − 3·29-s + 1.45·31-s − 7.80·32-s − 5.20·37-s + 2.14·38-s + 12.5·41-s − 8·43-s + 9.74·44-s + 1.16·46-s − 11.7·47-s + 0.543·49-s − 16.3·52-s − 5.58·53-s − 3.54·56-s + ⋯ |
| L(s) = 1 | + 1.51·2-s + 1.30·4-s − 1.03·7-s + 0.456·8-s + 1.12·11-s − 1.74·13-s − 1.57·14-s − 0.608·16-s + 0.229·19-s + 1.71·22-s + 0.113·23-s − 2.64·26-s − 1.35·28-s − 0.557·29-s + 0.261·31-s − 1.37·32-s − 0.855·37-s + 0.347·38-s + 1.95·41-s − 1.21·43-s + 1.46·44-s + 0.171·46-s − 1.71·47-s + 0.0776·49-s − 2.26·52-s − 0.766·53-s − 0.473·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.14T + 2T^{2} \) |
| 7 | \( 1 + 2.74T + 7T^{2} \) |
| 11 | \( 1 - 3.74T + 11T^{2} \) |
| 13 | \( 1 + 6.29T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 - 0.543T + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 1.45T + 31T^{2} \) |
| 37 | \( 1 + 5.20T + 37T^{2} \) |
| 41 | \( 1 - 12.5T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 + 5.58T + 53T^{2} \) |
| 59 | \( 1 + 5.25T + 59T^{2} \) |
| 61 | \( 1 + 8.49T + 61T^{2} \) |
| 67 | \( 1 - 2.83T + 67T^{2} \) |
| 71 | \( 1 + 7.83T + 71T^{2} \) |
| 73 | \( 1 + 7.58T + 73T^{2} \) |
| 79 | \( 1 + 7.52T + 79T^{2} \) |
| 83 | \( 1 + 2.25T + 83T^{2} \) |
| 89 | \( 1 + 4.49T + 89T^{2} \) |
| 97 | \( 1 - 16.8T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.66300730923849262192249852102, −6.97739169305391262865488561382, −6.41946451270437570374021085728, −5.77134607730961646704954905063, −4.86943604482293625300730900714, −4.34431192223846829853141256270, −3.37470672264428784343820599371, −2.91127296890655615025044537210, −1.81517967776236857049437805209, 0,
1.81517967776236857049437805209, 2.91127296890655615025044537210, 3.37470672264428784343820599371, 4.34431192223846829853141256270, 4.86943604482293625300730900714, 5.77134607730961646704954905063, 6.41946451270437570374021085728, 6.97739169305391262865488561382, 7.66300730923849262192249852102
