| L(s) = 1 | − 2·2-s + 2·4-s − 7-s − 2·13-s + 2·14-s − 4·16-s + 3·17-s + 5·19-s + 3·23-s + 4·26-s − 2·28-s − 3·29-s + 8·32-s − 6·34-s + 6·37-s − 10·38-s − 4·41-s + 7·43-s − 6·46-s − 4·47-s + 49-s − 4·52-s + 9·53-s + 6·58-s + 11·59-s + 61-s − 8·64-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.377·7-s − 0.554·13-s + 0.534·14-s − 16-s + 0.727·17-s + 1.14·19-s + 0.625·23-s + 0.784·26-s − 0.377·28-s − 0.557·29-s + 1.41·32-s − 1.02·34-s + 0.986·37-s − 1.62·38-s − 0.624·41-s + 1.06·43-s − 0.884·46-s − 0.583·47-s + 1/7·49-s − 0.554·52-s + 1.23·53-s + 0.787·58-s + 1.43·59-s + 0.128·61-s − 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−13.25473326401763, −12.85273236778148, −12.32742018145276, −11.72916445854089, −11.34080171399777, −10.97314917548028, −10.23027872125786, −9.915414994214504, −9.658107506138402, −9.125208067773947, −8.649905809313614, −8.168870887249135, −7.606153027489308, −7.226703058424396, −6.941418392196251, −6.180980165325497, −5.596947164287128, −5.120476817114116, −4.503432931228932, −3.795902102384801, −3.231546727426530, −2.547950341571887, −2.078322753950209, −1.145597648550260, −0.8530214744117783, 0,
0.8530214744117783, 1.145597648550260, 2.078322753950209, 2.547950341571887, 3.231546727426530, 3.795902102384801, 4.503432931228932, 5.120476817114116, 5.596947164287128, 6.180980165325497, 6.941418392196251, 7.226703058424396, 7.606153027489308, 8.168870887249135, 8.649905809313614, 9.125208067773947, 9.658107506138402, 9.915414994214504, 10.23027872125786, 10.97314917548028, 11.34080171399777, 11.72916445854089, 12.32742018145276, 12.85273236778148, 13.25473326401763
