| L(s) = 1 | + 1.76·2-s + 0.991·3-s + 1.12·4-s + 1.75·6-s − 7-s − 1.55·8-s − 2.01·9-s + 3.44·11-s + 1.11·12-s + 1.76·13-s − 1.76·14-s − 4.98·16-s − 3.70·17-s − 3.56·18-s − 8.16·19-s − 0.991·21-s + 6.08·22-s + 23-s − 1.53·24-s + 3.12·26-s − 4.97·27-s − 1.12·28-s + 1.63·29-s − 6.49·31-s − 5.70·32-s + 3.41·33-s − 6.54·34-s + ⋯ |
| L(s) = 1 | + 1.24·2-s + 0.572·3-s + 0.560·4-s + 0.714·6-s − 0.377·7-s − 0.549·8-s − 0.672·9-s + 1.03·11-s + 0.320·12-s + 0.490·13-s − 0.472·14-s − 1.24·16-s − 0.898·17-s − 0.839·18-s − 1.87·19-s − 0.216·21-s + 1.29·22-s + 0.208·23-s − 0.314·24-s + 0.612·26-s − 0.957·27-s − 0.211·28-s + 0.303·29-s − 1.16·31-s − 1.00·32-s + 0.594·33-s − 1.12·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 1.76T + 2T^{2} \) |
| 3 | \( 1 - 0.991T + 3T^{2} \) |
| 11 | \( 1 - 3.44T + 11T^{2} \) |
| 13 | \( 1 - 1.76T + 13T^{2} \) |
| 17 | \( 1 + 3.70T + 17T^{2} \) |
| 19 | \( 1 + 8.16T + 19T^{2} \) |
| 29 | \( 1 - 1.63T + 29T^{2} \) |
| 31 | \( 1 + 6.49T + 31T^{2} \) |
| 37 | \( 1 + 1.57T + 37T^{2} \) |
| 41 | \( 1 - 5.67T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 + 8.58T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 + 0.498T + 59T^{2} \) |
| 61 | \( 1 - 4.17T + 61T^{2} \) |
| 67 | \( 1 + 0.0800T + 67T^{2} \) |
| 71 | \( 1 + 9.21T + 71T^{2} \) |
| 73 | \( 1 - 16.3T + 73T^{2} \) |
| 79 | \( 1 + 4.09T + 79T^{2} \) |
| 83 | \( 1 + 14.8T + 83T^{2} \) |
| 89 | \( 1 + 7.81T + 89T^{2} \) |
| 97 | \( 1 - 4.40T + 97T^{2} \) |
| 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
−8.320013050680306150903043230326, −6.97026752561498821874116414578, −6.45130420469169767285232363637, −5.87574406630437594454578676494, −4.94846331624984247102311536133, −4.04916099824444592847139221114, −3.65952632396751575494219378477, −2.73377309887323574482565063324, −1.89872996626120768674385202336, 0,
1.89872996626120768674385202336, 2.73377309887323574482565063324, 3.65952632396751575494219378477, 4.04916099824444592847139221114, 4.94846331624984247102311536133, 5.87574406630437594454578676494, 6.45130420469169767285232363637, 6.97026752561498821874116414578, 8.320013050680306150903043230326
