| L(s) = 1 | − 2·2-s + 2·3-s + 2·4-s − 2·5-s − 4·6-s + 2·9-s + 4·10-s − 2·11-s + 4·12-s + 6·13-s − 4·15-s − 4·16-s − 4·18-s − 6·19-s − 4·20-s + 4·22-s + 2·25-s − 12·26-s + 6·27-s + 10·29-s + 8·30-s + 8·32-s − 4·33-s + 4·36-s + 14·37-s + 12·38-s + 12·39-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 4-s − 0.894·5-s − 1.63·6-s + 2/3·9-s + 1.26·10-s − 0.603·11-s + 1.15·12-s + 1.66·13-s − 1.03·15-s − 16-s − 0.942·18-s − 1.37·19-s − 0.894·20-s + 0.852·22-s + 2/5·25-s − 2.35·26-s + 1.15·27-s + 1.85·29-s + 1.46·30-s + 1.41·32-s − 0.696·33-s + 2/3·36-s + 2.30·37-s + 1.94·38-s + 1.92·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 215296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 215296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.025912403\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.025912403\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 29 | $C_2$ | \( 1 - 10 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 178 T^{2} + p^{2} T^{4} \) |
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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
−11.17064486284309522043135500260, −10.49787030335791989380622559846, −10.27229539393092186529194551388, −10.14164603928217019603581386080, −9.086340322308936073433552815634, −8.923879498739120919958655551243, −8.488532525123689683358971969304, −8.418234991497091524088982635132, −7.71092027499623632358796634696, −7.58924605314444043080288836442, −6.87356743552435474321111196999, −6.31348360052828196418500160983, −5.96343891107981297675288425077, −4.75682438406371117726514066075, −4.39826627400591214273090228205, −3.89327055853524987976859397575, −2.99247340747472416724798512748, −2.66716144076510927078125996386, −1.68185685931309701192107479299, −0.78123862599408846071691935294,
0.78123862599408846071691935294, 1.68185685931309701192107479299, 2.66716144076510927078125996386, 2.99247340747472416724798512748, 3.89327055853524987976859397575, 4.39826627400591214273090228205, 4.75682438406371117726514066075, 5.96343891107981297675288425077, 6.31348360052828196418500160983, 6.87356743552435474321111196999, 7.58924605314444043080288836442, 7.71092027499623632358796634696, 8.418234991497091524088982635132, 8.488532525123689683358971969304, 8.923879498739120919958655551243, 9.086340322308936073433552815634, 10.14164603928217019603581386080, 10.27229539393092186529194551388, 10.49787030335791989380622559846, 11.17064486284309522043135500260
