| L(s) = 1 | − 32·2-s + 768·4-s − 1.63e4·8-s − 3.66e4·9-s + 1.87e4·11-s + 3.27e5·16-s + 1.17e6·18-s − 6.00e5·22-s − 1.95e6·23-s − 5.35e5·25-s − 8.63e6·29-s − 6.29e6·32-s − 2.81e7·36-s + 5.42e6·37-s − 6.95e7·43-s + 1.44e7·44-s + 6.26e7·46-s + 1.71e7·50-s + 1.36e8·53-s + 2.76e8·58-s + 1.17e8·64-s + 4.85e8·67-s − 1.88e8·71-s + 5.99e8·72-s − 1.73e8·74-s + 1.35e9·79-s + 9.53e8·81-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s − 1.86·9-s + 0.386·11-s + 5/4·16-s + 2.63·18-s − 0.546·22-s − 1.45·23-s − 0.273·25-s − 2.26·29-s − 1.06·32-s − 2.79·36-s + 0.476·37-s − 3.10·43-s + 0.579·44-s + 2.06·46-s + 0.387·50-s + 2.36·53-s + 3.20·58-s + 7/8·64-s + 2.94·67-s − 0.880·71-s + 2.63·72-s − 0.673·74-s + 3.91·79-s + 2.46·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.09813630754\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09813630754\) |
| \(L(\frac{11}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{4} T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 36614 T^{2} + p^{18} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 21402 p^{2} T^{2} + p^{18} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 9380 T + p^{9} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 10847863142 T^{2} + p^{18} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 227593410082 T^{2} + p^{18} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 328706803750 T^{2} + p^{18} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 978936 T + p^{9} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 4317214 T + p^{9} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 10714676730050 T^{2} + p^{18} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 73362 p T + p^{9} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 573721268650450 T^{2} + p^{18} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 34755692 T + p^{9} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 459315263762526 T^{2} + p^{18} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 68067926 T + p^{9} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 16154177553222390 T^{2} + p^{18} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 4112136386785286 T^{2} + p^{18} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 242944420 T + p^{9} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 94292464 T + p^{9} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 99965099610294034 T^{2} + p^{18} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 677625160 T + p^{9} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 179122938928854694 T^{2} + p^{18} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 392379595356141618 T^{2} + p^{18} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 89760987702549566 T^{2} + p^{18} T^{4} \) |
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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
−11.88957094057782725371145816238, −11.77731890479229605392023752249, −11.22709924927035041240914086657, −10.83696360152232826560446028902, −9.970244849056229712471789460756, −9.745359920034234528205641648291, −8.945259315550405161503235636837, −8.691130294232481270410143328926, −7.976399309074454096539369652185, −7.76157571024312546689075818226, −6.76825029591838927734460603952, −6.33808280467746762464433647702, −5.61819184868322243389641373546, −5.20782688521304806896483376777, −3.74422961774375350151341401908, −3.43001602999000888579882080514, −2.21196979229354497086175423702, −2.12902590130438814167142233980, −0.985486229266157748087357108733, −0.12036122011254254792983075852,
0.12036122011254254792983075852, 0.985486229266157748087357108733, 2.12902590130438814167142233980, 2.21196979229354497086175423702, 3.43001602999000888579882080514, 3.74422961774375350151341401908, 5.20782688521304806896483376777, 5.61819184868322243389641373546, 6.33808280467746762464433647702, 6.76825029591838927734460603952, 7.76157571024312546689075818226, 7.976399309074454096539369652185, 8.691130294232481270410143328926, 8.945259315550405161503235636837, 9.745359920034234528205641648291, 9.970244849056229712471789460756, 10.83696360152232826560446028902, 11.22709924927035041240914086657, 11.77731890479229605392023752249, 11.88957094057782725371145816238
