| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 4·13-s + 14-s + 16-s + 17-s + 6·19-s − 5·25-s − 4·26-s − 28-s + 6·29-s − 4·31-s − 32-s − 34-s − 7·37-s − 6·38-s + 9·41-s + 12·47-s + 49-s + 5·50-s + 4·52-s − 11·53-s + 56-s − 6·58-s + 59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s + 1.37·19-s − 25-s − 0.784·26-s − 0.188·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.171·34-s − 1.15·37-s − 0.973·38-s + 1.40·41-s + 1.75·47-s + 1/7·49-s + 0.707·50-s + 0.554·52-s − 1.51·53-s + 0.133·56-s − 0.787·58-s + 0.130·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 15 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.02107815051552, −12.42119465080590, −12.12226218590822, −11.63092517634237, −11.05959049741044, −10.78664705618447, −10.24794816079400, −9.648117371638439, −9.480601327287336, −8.834897649460283, −8.480585192262899, −7.902028613092955, −7.457475441513933, −7.065924829556055, −6.403539778828946, −6.025197686739023, −5.502180628536912, −5.070545552154845, −4.158258280758791, −3.749983397773615, −3.224187854165570, −2.659690731244236, −2.011911323811592, −1.249384230264041, −0.8827888850907985, 0,
0.8827888850907985, 1.249384230264041, 2.011911323811592, 2.659690731244236, 3.224187854165570, 3.749983397773615, 4.158258280758791, 5.070545552154845, 5.502180628536912, 6.025197686739023, 6.403539778828946, 7.065924829556055, 7.457475441513933, 7.902028613092955, 8.480585192262899, 8.834897649460283, 9.480601327287336, 9.648117371638439, 10.24794816079400, 10.78664705618447, 11.05959049741044, 11.63092517634237, 12.12226218590822, 12.42119465080590, 13.02107815051552
