L(s) = 1 | − 9-s − 8·13-s − 14·17-s − 4·37-s + 10·41-s + 6·49-s − 12·53-s − 20·61-s − 18·73-s − 8·81-s − 10·89-s + 4·97-s + 4·101-s − 12·109-s − 2·113-s + 8·117-s − 17·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 14·153-s + 157-s + 163-s + 167-s + ⋯ |
L(s) = 1 | − 1/3·9-s − 2.21·13-s − 3.39·17-s − 0.657·37-s + 1.56·41-s + 6/7·49-s − 1.64·53-s − 2.56·61-s − 2.10·73-s − 8/9·81-s − 1.05·89-s + 0.406·97-s + 0.398·101-s − 1.14·109-s − 0.188·113-s + 0.739·117-s − 1.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 1.13·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 129 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 138 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 41 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.134759772882302671014297333508, −9.019761262339603708999838080207, −8.502895825510286928619127625088, −7.991435171477414627374839974277, −7.48837173447236328840971748586, −7.27770992290665604131308317999, −6.79650445747981054968354064973, −6.48230954725936861904647715889, −5.97210425117937302891075442120, −5.55667119887767360405845391235, −4.85098271997603266871651021224, −4.50266260138179899244006162340, −4.49279084897985053201105520927, −3.75774139522476010258234381639, −2.80146563172608540635832310297, −2.67867745836366183057433831746, −2.16334873787357758717855430440, −1.53043952256460281858287612690, 0, 0,
1.53043952256460281858287612690, 2.16334873787357758717855430440, 2.67867745836366183057433831746, 2.80146563172608540635832310297, 3.75774139522476010258234381639, 4.49279084897985053201105520927, 4.50266260138179899244006162340, 4.85098271997603266871651021224, 5.55667119887767360405845391235, 5.97210425117937302891075442120, 6.48230954725936861904647715889, 6.79650445747981054968354064973, 7.27770992290665604131308317999, 7.48837173447236328840971748586, 7.991435171477414627374839974277, 8.502895825510286928619127625088, 9.019761262339603708999838080207, 9.134759772882302671014297333508