| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s + 3·11-s + 2·13-s + 16-s + 17-s + 2·19-s − 20-s − 3·22-s − 6·23-s − 4·25-s − 2·26-s + 3·29-s − 9·31-s − 32-s − 34-s + 6·37-s − 2·38-s + 40-s + 12·41-s − 12·43-s + 3·44-s + 6·46-s − 4·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s + 0.904·11-s + 0.554·13-s + 1/4·16-s + 0.242·17-s + 0.458·19-s − 0.223·20-s − 0.639·22-s − 1.25·23-s − 4/5·25-s − 0.392·26-s + 0.557·29-s − 1.61·31-s − 0.176·32-s − 0.171·34-s + 0.986·37-s − 0.324·38-s + 0.158·40-s + 1.87·41-s − 1.82·43-s + 0.452·44-s + 0.884·46-s − 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14994 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14994 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−16.30247629825328, −16.02469338049402, −15.30426275139747, −14.68291968857357, −14.28582401709830, −13.54672389733518, −12.96104244802624, −12.11951341802445, −11.82952852580489, −11.24514016721374, −10.72897570496886, −9.869317162348582, −9.590987423587666, −8.828797204566957, −8.321463725262989, −7.655286558581558, −7.233995056867029, −6.328503834080495, −5.967073323413883, −5.120761156073769, −3.997963516741908, −3.795044810283485, −2.781669874940770, −1.839236853205969, −1.102649150963550, 0,
1.102649150963550, 1.839236853205969, 2.781669874940770, 3.795044810283485, 3.997963516741908, 5.120761156073769, 5.967073323413883, 6.328503834080495, 7.233995056867029, 7.655286558581558, 8.321463725262989, 8.828797204566957, 9.590987423587666, 9.869317162348582, 10.72897570496886, 11.24514016721374, 11.82952852580489, 12.11951341802445, 12.96104244802624, 13.54672389733518, 14.28582401709830, 14.68291968857357, 15.30426275139747, 16.02469338049402, 16.30247629825328
