| L(s) = 1 | + 18·9-s + 32·11-s − 8·19-s + 484·29-s + 200·31-s − 276·41-s − 470·49-s + 536·59-s + 500·61-s − 1.70e3·71-s + 912·79-s − 405·81-s + 1.45e3·89-s + 576·99-s + 252·101-s − 52·109-s − 1.89e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.03e3·169-s + ⋯ |
| L(s) = 1 | + 2/3·9-s + 0.877·11-s − 0.0965·19-s + 3.09·29-s + 1.15·31-s − 1.05·41-s − 1.37·49-s + 1.18·59-s + 1.04·61-s − 2.84·71-s + 1.29·79-s − 5/9·81-s + 1.72·89-s + 0.584·99-s + 0.248·101-s − 0.0456·109-s − 1.42·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 0.468·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.785494615\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.785494615\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 p^{2} T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 470 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 16 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 1030 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 4926 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6378 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 242 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 100 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 90538 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 138 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 127330 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 207162 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 271510 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 268 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 250 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 423442 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 p T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 684398 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 456 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 955218 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 726 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 73538 T^{2} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.10443145592130948938608936754, −11.68686222968483056199168925348, −11.62044725708061388106079813757, −10.57496778737811968702947871957, −10.21226904755510500151509787291, −10.01842782216335593744034415460, −9.243279319903079200697319969872, −8.761268232543445528582876857554, −8.243939880456815058322571699039, −7.81041077003716212758398409145, −6.91494442855748265991722564748, −6.61140969717182928422524201413, −6.23302145187960794167270274872, −5.26242599014986401523644065854, −4.63788445674457340994818147276, −4.22196915722902203541349626735, −3.34046729813343879304697182523, −2.63600448645256205255636849807, −1.55783288810453490128570803726, −0.78880080919790876750296469868,
0.78880080919790876750296469868, 1.55783288810453490128570803726, 2.63600448645256205255636849807, 3.34046729813343879304697182523, 4.22196915722902203541349626735, 4.63788445674457340994818147276, 5.26242599014986401523644065854, 6.23302145187960794167270274872, 6.61140969717182928422524201413, 6.91494442855748265991722564748, 7.81041077003716212758398409145, 8.243939880456815058322571699039, 8.761268232543445528582876857554, 9.243279319903079200697319969872, 10.01842782216335593744034415460, 10.21226904755510500151509787291, 10.57496778737811968702947871957, 11.62044725708061388106079813757, 11.68686222968483056199168925348, 12.10443145592130948938608936754
