| L(s) = 1 | + 2.64·2-s + 5.00·4-s + 2.64·5-s + 7.93·8-s + 7.00·10-s − 2.64·11-s + 2·13-s + 11.0·16-s − 7·19-s + 13.2·20-s − 7.00·22-s − 7.93·23-s + 2.00·25-s + 5.29·26-s + 5.29·29-s − 3·31-s + 13.2·32-s − 3·37-s − 18.5·38-s + 21.0·40-s − 2.64·41-s + 8·43-s − 13.2·44-s − 21.0·46-s + 5.29·50-s + 10.0·52-s − 7.00·55-s + ⋯ |
| L(s) = 1 | + 1.87·2-s + 2.50·4-s + 1.18·5-s + 2.80·8-s + 2.21·10-s − 0.797·11-s + 0.554·13-s + 2.75·16-s − 1.60·19-s + 2.95·20-s − 1.49·22-s − 1.65·23-s + 0.400·25-s + 1.03·26-s + 0.982·29-s − 0.538·31-s + 2.33·32-s − 0.493·37-s − 3.00·38-s + 3.32·40-s − 0.413·41-s + 1.21·43-s − 1.99·44-s − 3.09·46-s + 0.748·50-s + 1.38·52-s − 0.943·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.046265697\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.046265697\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.64T + 2T^{2} \) |
| 5 | \( 1 - 2.64T + 5T^{2} \) |
| 11 | \( 1 + 2.64T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 7T + 19T^{2} \) |
| 23 | \( 1 + 7.93T + 23T^{2} \) |
| 29 | \( 1 - 5.29T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + 3T + 37T^{2} \) |
| 41 | \( 1 + 2.64T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 - 7.93T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 - 18.5T + 89T^{2} \) |
| 97 | \( 1 - 12T + 97T^{2} \) |
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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
−10.10298622004518825749087570464, −8.725418989418723089930402407623, −7.74443858311826301435970244497, −6.61972073495258015631834214141, −6.07355209842237419310298739082, −5.48769918369145578724784166699, −4.55752085290803469902278297907, −3.70742956508013399152369602234, −2.49286076782179912292461798588, −1.90129567862797964692328185767,
1.90129567862797964692328185767, 2.49286076782179912292461798588, 3.70742956508013399152369602234, 4.55752085290803469902278297907, 5.48769918369145578724784166699, 6.07355209842237419310298739082, 6.61972073495258015631834214141, 7.74443858311826301435970244497, 8.725418989418723089930402407623, 10.10298622004518825749087570464
