| L(s) = 1 | − 2·5-s + 9-s + 4·11-s − 25-s + 8·31-s + 8·37-s − 2·45-s − 16·47-s − 2·49-s + 24·53-s − 8·55-s − 8·59-s + 16·67-s + 16·71-s + 81-s − 20·89-s − 16·97-s + 4·99-s + 8·103-s + 8·113-s + 5·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1/3·9-s + 1.20·11-s − 1/5·25-s + 1.43·31-s + 1.31·37-s − 0.298·45-s − 2.33·47-s − 2/7·49-s + 3.29·53-s − 1.07·55-s − 1.04·59-s + 1.95·67-s + 1.89·71-s + 1/9·81-s − 2.11·89-s − 1.62·97-s + 0.402·99-s + 0.788·103-s + 0.752·113-s + 5/11·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.988972255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.988972255\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.026404297766136895662512055972, −7.25624151390069608510211286773, −7.03266227275545160492653105318, −6.62211565498966079669618967734, −6.20333685780982384479221491754, −5.73263529762399736386921427177, −5.14057785936848501579779029287, −4.60473178973771251911928934463, −4.23353884561558632655177150743, −3.81912715817204232545956328387, −3.41819111991102427617959030821, −2.72910793845063097667579181890, −2.11649935913762436171788765859, −1.30299406909277296839825968826, −0.65276342797331901274662808003,
0.65276342797331901274662808003, 1.30299406909277296839825968826, 2.11649935913762436171788765859, 2.72910793845063097667579181890, 3.41819111991102427617959030821, 3.81912715817204232545956328387, 4.23353884561558632655177150743, 4.60473178973771251911928934463, 5.14057785936848501579779029287, 5.73263529762399736386921427177, 6.20333685780982384479221491754, 6.62211565498966079669618967734, 7.03266227275545160492653105318, 7.25624151390069608510211286773, 8.026404297766136895662512055972
