| L(s) = 1 | + 81·3-s + 4.03e3·7-s + 6.56e3·9-s − 3.58e4·13-s − 2.58e5·19-s + 3.26e5·21-s + 3.90e5·25-s + 5.31e5·27-s − 1.80e6·31-s + 5.03e5·37-s − 2.90e6·39-s + 3.49e6·43-s + 1.05e7·49-s − 2.09e7·57-s − 2.38e7·61-s + 2.64e7·63-s − 5.42e6·67-s + 1.61e7·73-s + 3.16e7·75-s − 1.88e7·79-s + 4.30e7·81-s − 1.44e8·91-s − 1.46e8·93-s + 1.76e8·97-s + 4.44e7·103-s + 2.03e8·109-s + 4.07e7·111-s + ⋯ |
| L(s) = 1 | + 3-s + 1.68·7-s + 9-s − 1.25·13-s − 1.98·19-s + 1.68·21-s + 25-s + 27-s − 1.95·31-s + 0.268·37-s − 1.25·39-s + 1.02·43-s + 1.82·49-s − 1.98·57-s − 1.72·61-s + 1.68·63-s − 0.269·67-s + 0.569·73-s + 75-s − 0.484·79-s + 81-s − 2.10·91-s − 1.95·93-s + 1.99·97-s + 0.394·103-s + 1.43·109-s + 0.268·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.199441574\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.199441574\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{4} T \) |
| good | 5 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 7 | \( 1 - 4034 T + p^{8} T^{2} \) |
| 11 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 13 | \( 1 + 35806 T + p^{8} T^{2} \) |
| 17 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 19 | \( 1 + 258526 T + p^{8} T^{2} \) |
| 23 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 29 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 31 | \( 1 + 1809406 T + p^{8} T^{2} \) |
| 37 | \( 1 - 503522 T + p^{8} T^{2} \) |
| 41 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 43 | \( 1 - 3492194 T + p^{8} T^{2} \) |
| 47 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 53 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 59 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 61 | \( 1 + 23826526 T + p^{8} T^{2} \) |
| 67 | \( 1 + 5421406 T + p^{8} T^{2} \) |
| 71 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 73 | \( 1 - 16169282 T + p^{8} T^{2} \) |
| 79 | \( 1 + 18887038 T + p^{8} T^{2} \) |
| 83 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 89 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 97 | \( 1 - 176908034 T + p^{8} T^{2} \) |
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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
−18.39835047131318537675156112370, −17.02992374494271057873588988322, −14.96081803052509069010596064802, −14.41552812122034406883699247325, −12.63472183049208722041768024234, −10.75463369038939012050174186325, −8.835660323279886529745629569899, −7.52648925107158231173679842726, −4.58388500628342377229240116291, −2.05258098433685276040585832153,
2.05258098433685276040585832153, 4.58388500628342377229240116291, 7.52648925107158231173679842726, 8.835660323279886529745629569899, 10.75463369038939012050174186325, 12.63472183049208722041768024234, 14.41552812122034406883699247325, 14.96081803052509069010596064802, 17.02992374494271057873588988322, 18.39835047131318537675156112370
