| L(s) = 1 | − 2-s + 4-s + 5-s − 4·7-s − 8-s − 10-s − 6·11-s + 4·13-s + 4·14-s + 16-s − 6·17-s + 20-s + 6·22-s − 6·23-s + 25-s − 4·26-s − 4·28-s + 2·29-s − 32-s + 6·34-s − 4·35-s + 8·37-s − 40-s + 10·41-s − 4·43-s − 6·44-s + 6·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.51·7-s − 0.353·8-s − 0.316·10-s − 1.80·11-s + 1.10·13-s + 1.06·14-s + 1/4·16-s − 1.45·17-s + 0.223·20-s + 1.27·22-s − 1.25·23-s + 1/5·25-s − 0.784·26-s − 0.755·28-s + 0.371·29-s − 0.176·32-s + 1.02·34-s − 0.676·35-s + 1.31·37-s − 0.158·40-s + 1.56·41-s − 0.609·43-s − 0.904·44-s + 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−15.61777399078217, −15.07168366369945, −14.07128891714491, −13.56228126916050, −13.18816123852733, −12.75275514775079, −12.28988986333547, −11.25741600202732, −11.01558693260671, −10.34657941128534, −9.931578611235143, −9.530819370149893, −8.751758291524528, −8.447349710252230, −7.695303224000822, −7.142956461807685, −6.418109441359361, −6.005858199418084, −5.630294253453631, −4.574382986466275, −3.930024088031116, −3.020318287904218, −2.583993912786845, −1.965306754536642, −0.7486486953061625, 0,
0.7486486953061625, 1.965306754536642, 2.583993912786845, 3.020318287904218, 3.930024088031116, 4.574382986466275, 5.630294253453631, 6.005858199418084, 6.418109441359361, 7.142956461807685, 7.695303224000822, 8.447349710252230, 8.751758291524528, 9.530819370149893, 9.931578611235143, 10.34657941128534, 11.01558693260671, 11.25741600202732, 12.28988986333547, 12.75275514775079, 13.18816123852733, 13.56228126916050, 14.07128891714491, 15.07168366369945, 15.61777399078217
