| L(s) = 1 | + 3-s − 4·5-s + 7-s − 2·9-s − 2·13-s − 4·15-s − 5·17-s − 7·19-s + 21-s + 11·25-s − 5·27-s − 8·29-s − 2·31-s − 4·35-s + 8·37-s − 2·39-s − 2·41-s − 3·43-s + 8·45-s + 49-s − 5·51-s − 6·53-s − 7·57-s + 9·59-s + 6·61-s − 2·63-s + 8·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78·5-s + 0.377·7-s − 2/3·9-s − 0.554·13-s − 1.03·15-s − 1.21·17-s − 1.60·19-s + 0.218·21-s + 11/5·25-s − 0.962·27-s − 1.48·29-s − 0.359·31-s − 0.676·35-s + 1.31·37-s − 0.320·39-s − 0.312·41-s − 0.457·43-s + 1.19·45-s + 1/7·49-s − 0.700·51-s − 0.824·53-s − 0.927·57-s + 1.17·59-s + 0.768·61-s − 0.251·63-s + 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 17 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 13 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.99899407844726, −12.78589406059965, −12.15372631696678, −11.59903152736709, −11.30087678238648, −11.00917933414309, −10.57874558782377, −9.759861417362779, −9.232898274017929, −8.799172764433933, −8.307510551312660, −8.088469301572770, −7.621931769002018, −7.046441578623236, −6.651650607424321, −6.041836140314664, −5.242893995510662, −4.852554569591972, −4.109211769489709, −3.971173245758483, −3.441647568868886, −2.547199582675381, −2.394208910478221, −1.571222723305494, −0.4738242234869243, 0,
0.4738242234869243, 1.571222723305494, 2.394208910478221, 2.547199582675381, 3.441647568868886, 3.971173245758483, 4.109211769489709, 4.852554569591972, 5.242893995510662, 6.041836140314664, 6.651650607424321, 7.046441578623236, 7.621931769002018, 8.088469301572770, 8.307510551312660, 8.799172764433933, 9.232898274017929, 9.759861417362779, 10.57874558782377, 11.00917933414309, 11.30087678238648, 11.59903152736709, 12.15372631696678, 12.78589406059965, 12.99899407844726
