| L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s + 11-s + 13-s − 14-s + 16-s + 3·17-s − 7·19-s − 20-s + 22-s + 25-s + 26-s − 28-s + 3·29-s + 5·31-s + 32-s + 3·34-s + 35-s − 7·37-s − 7·38-s − 40-s − 10·43-s + 44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s + 0.301·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s − 1.60·19-s − 0.223·20-s + 0.213·22-s + 1/5·25-s + 0.196·26-s − 0.188·28-s + 0.557·29-s + 0.898·31-s + 0.176·32-s + 0.514·34-s + 0.169·35-s − 1.15·37-s − 1.13·38-s − 0.158·40-s − 1.52·43-s + 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−16.58076482658629, −15.90025790005115, −15.36431287859598, −14.77356133125973, −14.47280238264350, −13.61325316849603, −13.19866554010576, −12.64180476330824, −11.94375599810959, −11.70756292302109, −10.87535925701366, −10.27805067336060, −9.862628552425392, −8.820761046005883, −8.381885903604274, −7.768451949245052, −6.786951128440638, −6.572482308602210, −5.832332828989139, −4.973182736699687, −4.446023959396302, −3.623660665063453, −3.181981873904557, −2.210503768296105, −1.298751290964309, 0,
1.298751290964309, 2.210503768296105, 3.181981873904557, 3.623660665063453, 4.446023959396302, 4.973182736699687, 5.832332828989139, 6.572482308602210, 6.786951128440638, 7.768451949245052, 8.381885903604274, 8.820761046005883, 9.862628552425392, 10.27805067336060, 10.87535925701366, 11.70756292302109, 11.94375599810959, 12.64180476330824, 13.19866554010576, 13.61325316849603, 14.47280238264350, 14.77356133125973, 15.36431287859598, 15.90025790005115, 16.58076482658629
