| L(s) = 1 | − 2·2-s + 4·4-s − 3·5-s + 7·7-s − 8·8-s + 6·10-s + 11·11-s − 16·13-s − 14·14-s + 16·16-s − 6·17-s + 14·19-s − 12·20-s − 22·22-s + 51·23-s − 116·25-s + 32·26-s + 28·28-s − 54·29-s + 95·31-s − 32·32-s + 12·34-s − 21·35-s − 193·37-s − 28·38-s + 24·40-s − 102·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.268·5-s + 0.377·7-s − 0.353·8-s + 0.189·10-s + 0.301·11-s − 0.341·13-s − 0.267·14-s + 1/4·16-s − 0.0856·17-s + 0.169·19-s − 0.134·20-s − 0.213·22-s + 0.462·23-s − 0.927·25-s + 0.241·26-s + 0.188·28-s − 0.345·29-s + 0.550·31-s − 0.176·32-s + 0.0605·34-s − 0.101·35-s − 0.857·37-s − 0.119·38-s + 0.0948·40-s − 0.388·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p T \) |
| 11 | \( 1 - p T \) |
| good | 5 | \( 1 + 3 T + p^{3} T^{2} \) |
| 13 | \( 1 + 16 T + p^{3} T^{2} \) |
| 17 | \( 1 + 6 T + p^{3} T^{2} \) |
| 19 | \( 1 - 14 T + p^{3} T^{2} \) |
| 23 | \( 1 - 51 T + p^{3} T^{2} \) |
| 29 | \( 1 + 54 T + p^{3} T^{2} \) |
| 31 | \( 1 - 95 T + p^{3} T^{2} \) |
| 37 | \( 1 + 193 T + p^{3} T^{2} \) |
| 41 | \( 1 + 102 T + p^{3} T^{2} \) |
| 43 | \( 1 - 284 T + p^{3} T^{2} \) |
| 47 | \( 1 - 72 T + p^{3} T^{2} \) |
| 53 | \( 1 - 102 T + p^{3} T^{2} \) |
| 59 | \( 1 - 63 T + p^{3} T^{2} \) |
| 61 | \( 1 + 790 T + p^{3} T^{2} \) |
| 67 | \( 1 + 433 T + p^{3} T^{2} \) |
| 71 | \( 1 + 135 T + p^{3} T^{2} \) |
| 73 | \( 1 + 238 T + p^{3} T^{2} \) |
| 79 | \( 1 - 770 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1008 T + p^{3} T^{2} \) |
| 89 | \( 1 - 639 T + p^{3} T^{2} \) |
| 97 | \( 1 - 11 T + p^{3} T^{2} \) |
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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
−8.878794766988347207658259798136, −7.961485819829062135519775776640, −7.39424730246031550712372403459, −6.50421921073347384969529891281, −5.55484326742449531753816914370, −4.53486006758805221922261255434, −3.48463970006793038723866586532, −2.33071546155882399522375167369, −1.25184503462927494780642753189, 0,
1.25184503462927494780642753189, 2.33071546155882399522375167369, 3.48463970006793038723866586532, 4.53486006758805221922261255434, 5.55484326742449531753816914370, 6.50421921073347384969529891281, 7.39424730246031550712372403459, 7.961485819829062135519775776640, 8.878794766988347207658259798136
