| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s − 2·13-s − 15-s + 2·17-s − 4·19-s − 21-s − 23-s + 25-s − 27-s − 10·29-s − 8·31-s + 35-s + 6·37-s + 2·39-s + 2·41-s + 4·43-s + 45-s + 4·47-s + 49-s − 2·51-s + 6·53-s + 4·57-s + 4·59-s − 10·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.554·13-s − 0.258·15-s + 0.485·17-s − 0.917·19-s − 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s − 1.85·29-s − 1.43·31-s + 0.169·35-s + 0.986·37-s + 0.320·39-s + 0.312·41-s + 0.609·43-s + 0.149·45-s + 0.583·47-s + 1/7·49-s − 0.280·51-s + 0.824·53-s + 0.529·57-s + 0.520·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.40804692122339, −13.73729532888186, −13.18940472757607, −12.74478952599730, −12.37551732107203, −11.77317981401307, −11.22386975889377, −10.76191918322979, −10.47127275745160, −9.628622189335495, −9.388945685920175, −8.844368370710638, −8.032142208660701, −7.628294056344158, −7.104402168146815, −6.527169241987819, −5.812496328692635, −5.527701898730486, −5.020542547199375, −4.142456710405856, −3.933435709184747, −2.949421250174845, −2.167436415222972, −1.777920555629518, −0.8548326298546732, 0,
0.8548326298546732, 1.777920555629518, 2.167436415222972, 2.949421250174845, 3.933435709184747, 4.142456710405856, 5.020542547199375, 5.527701898730486, 5.812496328692635, 6.527169241987819, 7.104402168146815, 7.628294056344158, 8.032142208660701, 8.844368370710638, 9.388945685920175, 9.628622189335495, 10.47127275745160, 10.76191918322979, 11.22386975889377, 11.77317981401307, 12.37551732107203, 12.74478952599730, 13.18940472757607, 13.73729532888186, 14.40804692122339
