| L(s) = 1 | − 2-s − 4-s + 3·8-s − 9-s + 8·13-s − 16-s + 6·17-s + 18-s + 8·19-s + 6·25-s − 8·26-s − 5·32-s − 6·34-s + 36-s − 8·38-s + 16·43-s − 8·47-s − 6·49-s − 6·50-s − 8·52-s − 8·53-s + 7·64-s + 8·67-s − 6·68-s − 3·72-s − 8·76-s + 81-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s − 1/3·9-s + 2.21·13-s − 1/4·16-s + 1.45·17-s + 0.235·18-s + 1.83·19-s + 6/5·25-s − 1.56·26-s − 0.883·32-s − 1.02·34-s + 1/6·36-s − 1.29·38-s + 2.43·43-s − 1.16·47-s − 6/7·49-s − 0.848·50-s − 1.10·52-s − 1.09·53-s + 7/8·64-s + 0.977·67-s − 0.727·68-s − 0.353·72-s − 0.917·76-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.255038437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.255038437\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 17 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−9.263844194681166114188367593089, −8.786064376164005336469716390831, −8.187647726261926704801642466388, −7.980228706890142293229457964473, −7.52743549649696816089268395025, −6.83319779824037045637920692237, −6.30099297455096275377142846127, −5.58979960708130542984442121090, −5.37983948625457544432631424141, −4.66316115748338781409691276674, −3.86444939526287277700079018836, −3.43995761522107744864665143089, −2.84320179983606393123498192769, −1.35741193497302407292590987356, −1.04486348591446020583807845079,
1.04486348591446020583807845079, 1.35741193497302407292590987356, 2.84320179983606393123498192769, 3.43995761522107744864665143089, 3.86444939526287277700079018836, 4.66316115748338781409691276674, 5.37983948625457544432631424141, 5.58979960708130542984442121090, 6.30099297455096275377142846127, 6.83319779824037045637920692237, 7.52743549649696816089268395025, 7.980228706890142293229457964473, 8.187647726261926704801642466388, 8.786064376164005336469716390831, 9.263844194681166114188367593089
