| L(s) = 1 | + 6·3-s − 5·5-s + 34·7-s + 9·9-s − 16·11-s + 58·13-s − 30·15-s − 70·17-s − 4·19-s + 204·21-s + 134·23-s + 25·25-s − 108·27-s − 242·29-s − 100·31-s − 96·33-s − 170·35-s − 438·37-s + 348·39-s − 138·41-s − 178·43-s − 45·45-s − 22·47-s + 813·49-s − 420·51-s + 162·53-s + 80·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 1.83·7-s + 1/3·9-s − 0.438·11-s + 1.23·13-s − 0.516·15-s − 0.998·17-s − 0.0482·19-s + 2.11·21-s + 1.21·23-s + 1/5·25-s − 0.769·27-s − 1.54·29-s − 0.579·31-s − 0.506·33-s − 0.821·35-s − 1.94·37-s + 1.42·39-s − 0.525·41-s − 0.631·43-s − 0.149·45-s − 0.0682·47-s + 2.37·49-s − 1.15·51-s + 0.419·53-s + 0.196·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.199074385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.199074385\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + p T \) |
| good | 3 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 7 | \( 1 - 34 T + p^{3} T^{2} \) |
| 11 | \( 1 + 16 T + p^{3} T^{2} \) |
| 13 | \( 1 - 58 T + p^{3} T^{2} \) |
| 17 | \( 1 + 70 T + p^{3} T^{2} \) |
| 19 | \( 1 + 4 T + p^{3} T^{2} \) |
| 23 | \( 1 - 134 T + p^{3} T^{2} \) |
| 29 | \( 1 + 242 T + p^{3} T^{2} \) |
| 31 | \( 1 + 100 T + p^{3} T^{2} \) |
| 37 | \( 1 + 438 T + p^{3} T^{2} \) |
| 41 | \( 1 + 138 T + p^{3} T^{2} \) |
| 43 | \( 1 + 178 T + p^{3} T^{2} \) |
| 47 | \( 1 + 22 T + p^{3} T^{2} \) |
| 53 | \( 1 - 162 T + p^{3} T^{2} \) |
| 59 | \( 1 - 268 T + p^{3} T^{2} \) |
| 61 | \( 1 - 250 T + p^{3} T^{2} \) |
| 67 | \( 1 + 422 T + p^{3} T^{2} \) |
| 71 | \( 1 - 12 p T + p^{3} T^{2} \) |
| 73 | \( 1 - 306 T + p^{3} T^{2} \) |
| 79 | \( 1 - 456 T + p^{3} T^{2} \) |
| 83 | \( 1 + 434 T + p^{3} T^{2} \) |
| 89 | \( 1 + 726 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1378 T + p^{3} T^{2} \) |
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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
−13.98553828062650598045183884108, −13.14064112858471889293621920579, −11.45901283928387784939541881848, −10.84239329356375622924505266728, −8.863043885947949707333032492577, −8.389003226449925099159889894829, −7.30489249990647196376005160239, −5.16709059318517025165599306336, −3.69192496010823120486146504641, −1.88698110066896039804741983862,
1.88698110066896039804741983862, 3.69192496010823120486146504641, 5.16709059318517025165599306336, 7.30489249990647196376005160239, 8.389003226449925099159889894829, 8.863043885947949707333032492577, 10.84239329356375622924505266728, 11.45901283928387784939541881848, 13.14064112858471889293621920579, 13.98553828062650598045183884108
