| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 3·7-s − 8-s − 2·9-s + 10-s + 3·11-s + 12-s + 13-s − 3·14-s − 15-s + 16-s + 2·17-s + 2·18-s − 6·19-s − 20-s + 3·21-s − 3·22-s + 2·23-s − 24-s − 4·25-s − 26-s − 5·27-s + 3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 1.13·7-s − 0.353·8-s − 2/3·9-s + 0.316·10-s + 0.904·11-s + 0.288·12-s + 0.277·13-s − 0.801·14-s − 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.471·18-s − 1.37·19-s − 0.223·20-s + 0.654·21-s − 0.639·22-s + 0.417·23-s − 0.204·24-s − 4/5·25-s − 0.196·26-s − 0.962·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30446 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30446 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 1171 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 9 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−15.17242247161577, −14.81322714763631, −14.59336062242465, −13.90847646666377, −13.42163692458760, −12.60314849092854, −11.99420178494197, −11.54039226772436, −11.16652836812789, −10.61232529609980, −9.965256901940755, −9.132419151680470, −8.935317833034313, −8.220571998085814, −7.955595752018150, −7.442631216904913, −6.572296280532948, −6.071711355387371, −5.408554559355164, −4.405890945971702, −4.105909236040091, −3.189374324767286, −2.544383523125871, −1.759562510469588, −1.127077812938783, 0,
1.127077812938783, 1.759562510469588, 2.544383523125871, 3.189374324767286, 4.105909236040091, 4.405890945971702, 5.408554559355164, 6.071711355387371, 6.572296280532948, 7.442631216904913, 7.955595752018150, 8.220571998085814, 8.935317833034313, 9.132419151680470, 9.965256901940755, 10.61232529609980, 11.16652836812789, 11.54039226772436, 11.99420178494197, 12.60314849092854, 13.42163692458760, 13.90847646666377, 14.59336062242465, 14.81322714763631, 15.17242247161577
