| L(s) = 1 | − 2·5-s − 9-s + 2·13-s − 2·17-s + 8·19-s − 8·23-s − 3·25-s − 10·37-s + 2·45-s − 5·49-s + 8·59-s − 4·65-s − 8·79-s − 8·81-s + 8·83-s + 4·85-s − 16·95-s + 8·103-s − 18·109-s + 4·113-s + 16·115-s − 2·117-s − 22·121-s + 10·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1/3·9-s + 0.554·13-s − 0.485·17-s + 1.83·19-s − 1.66·23-s − 3/5·25-s − 1.64·37-s + 0.298·45-s − 5/7·49-s + 1.04·59-s − 0.496·65-s − 0.900·79-s − 8/9·81-s + 0.878·83-s + 0.433·85-s − 1.64·95-s + 0.788·103-s − 1.72·109-s + 0.376·113-s + 1.49·115-s − 0.184·117-s − 2·121-s + 0.894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 63 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−8.875695304099684896281182707766, −8.420945047513015779175356935662, −7.935387512355577643087283909151, −7.65104784567127020197799894934, −7.06455621115253245281313072600, −6.56480351600023000949595423956, −5.91753095980449469480233466188, −5.47293856182412718696639704946, −4.93230086189046741029681749520, −4.13991592642427677936275988561, −3.71352235547140785875774871076, −3.24216290472028332104415975278, −2.33989940612046940156518447462, −1.40740953620497473077259660886, 0,
1.40740953620497473077259660886, 2.33989940612046940156518447462, 3.24216290472028332104415975278, 3.71352235547140785875774871076, 4.13991592642427677936275988561, 4.93230086189046741029681749520, 5.47293856182412718696639704946, 5.91753095980449469480233466188, 6.56480351600023000949595423956, 7.06455621115253245281313072600, 7.65104784567127020197799894934, 7.935387512355577643087283909151, 8.420945047513015779175356935662, 8.875695304099684896281182707766
