| L(s) = 1 | − 9·3-s − 92·5-s + 26·7-s + 81·9-s − 121·11-s − 692·13-s + 828·15-s − 1.44e3·17-s − 2.16e3·19-s − 234·21-s + 1.58e3·23-s + 5.33e3·25-s − 729·27-s − 5.52e3·29-s − 4.79e3·31-s + 1.08e3·33-s − 2.39e3·35-s − 1.01e4·37-s + 6.22e3·39-s − 1.06e4·41-s − 8.58e3·43-s − 7.45e3·45-s + 2.36e3·47-s − 1.61e4·49-s + 1.29e4·51-s − 3.08e4·53-s + 1.11e4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.64·5-s + 0.200·7-s + 1/3·9-s − 0.301·11-s − 1.13·13-s + 0.950·15-s − 1.21·17-s − 1.37·19-s − 0.115·21-s + 0.623·23-s + 1.70·25-s − 0.192·27-s − 1.22·29-s − 0.895·31-s + 0.174·33-s − 0.330·35-s − 1.22·37-s + 0.655·39-s − 0.986·41-s − 0.707·43-s − 0.548·45-s + 0.155·47-s − 0.959·49-s + 0.698·51-s − 1.50·53-s + 0.496·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.04777051678\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04777051678\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{2} T \) |
| 11 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 + 92 T + p^{5} T^{2} \) |
| 7 | \( 1 - 26 T + p^{5} T^{2} \) |
| 13 | \( 1 + 692 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1442 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2160 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1582 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5526 T + p^{5} T^{2} \) |
| 31 | \( 1 + 4792 T + p^{5} T^{2} \) |
| 37 | \( 1 + 10194 T + p^{5} T^{2} \) |
| 41 | \( 1 + 10622 T + p^{5} T^{2} \) |
| 43 | \( 1 + 8580 T + p^{5} T^{2} \) |
| 47 | \( 1 - 2362 T + p^{5} T^{2} \) |
| 53 | \( 1 + 30804 T + p^{5} T^{2} \) |
| 59 | \( 1 + 6416 T + p^{5} T^{2} \) |
| 61 | \( 1 - 42096 T + p^{5} T^{2} \) |
| 67 | \( 1 - 28444 T + p^{5} T^{2} \) |
| 71 | \( 1 + 45690 T + p^{5} T^{2} \) |
| 73 | \( 1 + 18374 T + p^{5} T^{2} \) |
| 79 | \( 1 - 105214 T + p^{5} T^{2} \) |
| 83 | \( 1 + 62292 T + p^{5} T^{2} \) |
| 89 | \( 1 + 72246 T + p^{5} T^{2} \) |
| 97 | \( 1 - 79262 T + p^{5} T^{2} \) |
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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
−10.32226873039795482108323631095, −9.032638182974282806040469232983, −8.186385489007460685630941832315, −7.29568178558135056872132080754, −6.65956216119997440458053870166, −5.13277175495643301209362276863, −4.46470924023044954617225422882, −3.46977105541894737896900561917, −1.98545404471747681032946361297, −0.10612555623969945776101962116,
0.10612555623969945776101962116, 1.98545404471747681032946361297, 3.46977105541894737896900561917, 4.46470924023044954617225422882, 5.13277175495643301209362276863, 6.65956216119997440458053870166, 7.29568178558135056872132080754, 8.186385489007460685630941832315, 9.032638182974282806040469232983, 10.32226873039795482108323631095
