| L(s) = 1 | − 4·7-s − 14·13-s − 8·19-s + 5·25-s − 11·31-s + 37-s − 26·43-s + 9·49-s + 61-s − 11·67-s + 10·73-s + 13·79-s + 56·91-s − 38·97-s + 7·103-s + 19·109-s + 11·121-s + 127-s + 131-s + 32·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 3.88·13-s − 1.83·19-s + 25-s − 1.97·31-s + 0.164·37-s − 3.96·43-s + 9/7·49-s + 0.128·61-s − 1.34·67-s + 1.17·73-s + 1.46·79-s + 5.87·91-s − 3.85·97-s + 0.689·103-s + 1.81·109-s + 121-s + 0.0887·127-s + 0.0873·131-s + 2.77·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 19 T + p T^{2} )^{2} \) |
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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
−9.904687386031125116896321677541, −9.887322470137963680809190392534, −9.291735066432182844123465416347, −9.120512558456089903221012837588, −8.381092038084311284602853670839, −8.001563899498960574820874494160, −7.32659321742139427561074640349, −7.06476342203788797954215674069, −6.64877547330098715349704762108, −6.46392234880406541068158958085, −5.41228734321090273550459107508, −5.30836994320640864202856382895, −4.62216698753173184103127266406, −4.32295978299945350176159014135, −3.33131925600993062102943193966, −3.09513733736063924369718113462, −2.26555039051722170913284208592, −2.01116051270762943663652030735, 0, 0,
2.01116051270762943663652030735, 2.26555039051722170913284208592, 3.09513733736063924369718113462, 3.33131925600993062102943193966, 4.32295978299945350176159014135, 4.62216698753173184103127266406, 5.30836994320640864202856382895, 5.41228734321090273550459107508, 6.46392234880406541068158958085, 6.64877547330098715349704762108, 7.06476342203788797954215674069, 7.32659321742139427561074640349, 8.001563899498960574820874494160, 8.381092038084311284602853670839, 9.120512558456089903221012837588, 9.291735066432182844123465416347, 9.887322470137963680809190392534, 9.904687386031125116896321677541
