| L(s) = 1 | + 2-s + 4-s + 8-s + 2·9-s + 16-s + 2·18-s + 10·25-s − 4·29-s + 32-s + 2·36-s − 4·37-s − 7·49-s + 10·50-s + 28·53-s − 4·58-s + 64-s + 2·72-s − 4·74-s − 5·81-s − 7·98-s + 10·100-s + 28·106-s − 4·109-s + 4·113-s − 4·116-s + 121-s + 127-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 2/3·9-s + 1/4·16-s + 0.471·18-s + 2·25-s − 0.742·29-s + 0.176·32-s + 1/3·36-s − 0.657·37-s − 49-s + 1.41·50-s + 3.84·53-s − 0.525·58-s + 1/8·64-s + 0.235·72-s − 0.464·74-s − 5/9·81-s − 0.707·98-s + 100-s + 2.71·106-s − 0.383·109-s + 0.376·113-s − 0.371·116-s + 1/11·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189728 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189728 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.892774765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.892774765\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−9.138355316340654241373080875063, −8.550966502145525084864492626185, −8.282024530209696904037410734491, −7.44090205001602193090431329744, −7.07845764333392661979449060345, −6.84693669503363575874611229048, −6.15586689609603382481791658368, −5.55424893264437314769560336816, −5.13419306901755155276020089858, −4.55978687407873991200208708116, −4.01880193632796957500043528001, −3.43901875101765770181944718211, −2.75386392283221271014365340678, −2.01705469370341431396968543751, −1.05865158925595700954317475021,
1.05865158925595700954317475021, 2.01705469370341431396968543751, 2.75386392283221271014365340678, 3.43901875101765770181944718211, 4.01880193632796957500043528001, 4.55978687407873991200208708116, 5.13419306901755155276020089858, 5.55424893264437314769560336816, 6.15586689609603382481791658368, 6.84693669503363575874611229048, 7.07845764333392661979449060345, 7.44090205001602193090431329744, 8.282024530209696904037410734491, 8.550966502145525084864492626185, 9.138355316340654241373080875063
