| L(s) = 1 | + 3-s + 2·5-s + 9-s − 11-s − 13-s + 2·15-s − 6·17-s − 4·19-s + 8·23-s − 25-s + 27-s + 10·29-s − 33-s − 6·37-s − 39-s + 10·41-s + 4·43-s + 2·45-s − 8·47-s − 7·49-s − 6·51-s + 10·53-s − 2·55-s − 4·57-s − 12·59-s − 14·61-s − 2·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s − 0.301·11-s − 0.277·13-s + 0.516·15-s − 1.45·17-s − 0.917·19-s + 1.66·23-s − 1/5·25-s + 0.192·27-s + 1.85·29-s − 0.174·33-s − 0.986·37-s − 0.160·39-s + 1.56·41-s + 0.609·43-s + 0.298·45-s − 1.16·47-s − 49-s − 0.840·51-s + 1.37·53-s − 0.269·55-s − 0.529·57-s − 1.56·59-s − 1.79·61-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−15.39328305384460, −14.97439371571647, −14.47011360690890, −13.73911312486202, −13.53161864310171, −12.95198832701130, −12.49785367177328, −11.83709738315092, −10.96784383244205, −10.64033733421086, −10.15952865214403, −9.276379611415216, −9.117262637398336, −8.514375942582993, −7.859508020101338, −7.154909270000043, −6.547660285945114, −6.166277099910205, −5.297660718592769, −4.597349453542658, −4.279477473991748, −3.059855834341203, −2.706115780331364, −1.982782650885915, −1.255601611586146, 0,
1.255601611586146, 1.982782650885915, 2.706115780331364, 3.059855834341203, 4.279477473991748, 4.597349453542658, 5.297660718592769, 6.166277099910205, 6.547660285945114, 7.154909270000043, 7.859508020101338, 8.514375942582993, 9.117262637398336, 9.276379611415216, 10.15952865214403, 10.64033733421086, 10.96784383244205, 11.83709738315092, 12.49785367177328, 12.95198832701130, 13.53161864310171, 13.73911312486202, 14.47011360690890, 14.97439371571647, 15.39328305384460
