| L(s) = 1 | − 5·5-s − 36·7-s − 64·11-s − 65·13-s + 59·17-s + 28·19-s − 160·23-s − 100·25-s − 57·29-s − 164·31-s + 180·35-s − 321·37-s − 246·41-s + 8·43-s − 84·47-s + 953·49-s + 478·53-s + 320·55-s + 32·59-s + 415·61-s + 325·65-s + 220·67-s − 884·71-s − 77·73-s + 2.30e3·77-s + 80·79-s − 1.26e3·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.94·7-s − 1.75·11-s − 1.38·13-s + 0.841·17-s + 0.338·19-s − 1.45·23-s − 4/5·25-s − 0.364·29-s − 0.950·31-s + 0.869·35-s − 1.42·37-s − 0.937·41-s + 0.0283·43-s − 0.260·47-s + 2.77·49-s + 1.23·53-s + 0.784·55-s + 0.0706·59-s + 0.871·61-s + 0.620·65-s + 0.401·67-s − 1.47·71-s − 0.123·73-s + 3.40·77-s + 0.113·79-s − 1.67·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + p T + p^{3} T^{2} \) |
| 7 | \( 1 + 36 T + p^{3} T^{2} \) |
| 11 | \( 1 + 64 T + p^{3} T^{2} \) |
| 13 | \( 1 + 5 p T + p^{3} T^{2} \) |
| 17 | \( 1 - 59 T + p^{3} T^{2} \) |
| 19 | \( 1 - 28 T + p^{3} T^{2} \) |
| 23 | \( 1 + 160 T + p^{3} T^{2} \) |
| 29 | \( 1 + 57 T + p^{3} T^{2} \) |
| 31 | \( 1 + 164 T + p^{3} T^{2} \) |
| 37 | \( 1 + 321 T + p^{3} T^{2} \) |
| 41 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 43 | \( 1 - 8 T + p^{3} T^{2} \) |
| 47 | \( 1 + 84 T + p^{3} T^{2} \) |
| 53 | \( 1 - 478 T + p^{3} T^{2} \) |
| 59 | \( 1 - 32 T + p^{3} T^{2} \) |
| 61 | \( 1 - 415 T + p^{3} T^{2} \) |
| 67 | \( 1 - 220 T + p^{3} T^{2} \) |
| 71 | \( 1 + 884 T + p^{3} T^{2} \) |
| 73 | \( 1 + 77 T + p^{3} T^{2} \) |
| 79 | \( 1 - 80 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1268 T + p^{3} T^{2} \) |
| 89 | \( 1 - 123 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1346 T + p^{3} T^{2} \) |
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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
−8.386370699478427411626079974671, −7.49581671374368142921969406285, −7.04697343954147698993059590694, −5.80752325448884665852690740723, −5.27730032127781935077654260688, −3.89110296331181546343796825061, −3.12389512869914906158175010770, −2.25472236188474290818492346813, 0, 0,
2.25472236188474290818492346813, 3.12389512869914906158175010770, 3.89110296331181546343796825061, 5.27730032127781935077654260688, 5.80752325448884665852690740723, 7.04697343954147698993059590694, 7.49581671374368142921969406285, 8.386370699478427411626079974671
