| L(s) = 1 | − 2·3-s − 2·5-s + 3·9-s + 4·15-s + 12·17-s + 8·19-s + 3·25-s − 4·27-s + 16·31-s − 6·45-s + 2·49-s − 24·51-s − 16·57-s + 24·59-s − 4·61-s − 8·67-s + 20·73-s − 6·75-s − 16·79-s + 5·81-s − 24·85-s − 32·93-s − 16·95-s − 12·101-s + 32·103-s + 24·107-s + 10·121-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 9-s + 1.03·15-s + 2.91·17-s + 1.83·19-s + 3/5·25-s − 0.769·27-s + 2.87·31-s − 0.894·45-s + 2/7·49-s − 3.36·51-s − 2.11·57-s + 3.12·59-s − 0.512·61-s − 0.977·67-s + 2.34·73-s − 0.692·75-s − 1.80·79-s + 5/9·81-s − 2.60·85-s − 3.31·93-s − 1.64·95-s − 1.19·101-s + 3.15·103-s + 2.32·107-s + 0.909·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20793600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20793600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.270526445\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.270526445\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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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
−8.460099683441095365850628956192, −7.953257399028814320626868646625, −7.71766173771568017054760008637, −7.47102454236229231824003491264, −7.16214960691537572797958656620, −6.68424617436129606578978649668, −6.23025195634574107438497650884, −6.00145383678918749493074336476, −5.44738834594586071029387711032, −5.12980018325680849356282657480, −5.10017814061457224280113225587, −4.40883707682663410609884025526, −4.03166951358444206268067082336, −3.57711168333186923568351602993, −3.15227481519785940915430120284, −2.91332809189280350639626598764, −2.17246293991768075832821301094, −1.14846509315689869643883125877, −1.07474959731898934749602728327, −0.61922586406574319351608722580,
0.61922586406574319351608722580, 1.07474959731898934749602728327, 1.14846509315689869643883125877, 2.17246293991768075832821301094, 2.91332809189280350639626598764, 3.15227481519785940915430120284, 3.57711168333186923568351602993, 4.03166951358444206268067082336, 4.40883707682663410609884025526, 5.10017814061457224280113225587, 5.12980018325680849356282657480, 5.44738834594586071029387711032, 6.00145383678918749493074336476, 6.23025195634574107438497650884, 6.68424617436129606578978649668, 7.16214960691537572797958656620, 7.47102454236229231824003491264, 7.71766173771568017054760008637, 7.953257399028814320626868646625, 8.460099683441095365850628956192
