| L(s) = 1 | − 3·3-s − 6·5-s + 6·9-s + 18·15-s − 6·17-s + 17·25-s − 9·27-s + 14·37-s − 12·41-s − 8·43-s − 36·45-s + 6·47-s + 18·51-s − 6·59-s − 10·67-s − 51·75-s + 2·79-s + 9·81-s − 24·83-s + 36·85-s + 18·89-s + 18·101-s − 34·109-s − 42·111-s − 5·121-s + 36·123-s − 18·125-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 2.68·5-s + 2·9-s + 4.64·15-s − 1.45·17-s + 17/5·25-s − 1.73·27-s + 2.30·37-s − 1.87·41-s − 1.21·43-s − 5.36·45-s + 0.875·47-s + 2.52·51-s − 0.781·59-s − 1.22·67-s − 5.88·75-s + 0.225·79-s + 81-s − 2.63·83-s + 3.90·85-s + 1.90·89-s + 1.79·101-s − 3.25·109-s − 3.98·111-s − 0.454·121-s + 3.24·123-s − 1.60·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 79 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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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
−8.846627195615937834471654949274, −8.260881621114192979309572123975, −7.76067301561830944874172980745, −7.74965789236081186774395995966, −7.27062063980197453505704949253, −6.77760590509000362233881941072, −6.50001199665982583334120459203, −6.28367676954311908126695486635, −5.47522858693247873179393504889, −5.28179898313075499340992032338, −4.51024131722509948981894231522, −4.45026094027893276220404624101, −4.20950183504054508342572519817, −3.62627962375111411256766350452, −3.18093684708761575735433762568, −2.50368180196714403235931239629, −1.60956179076699655136018697458, −0.879943412796931761594590900225, 0, 0,
0.879943412796931761594590900225, 1.60956179076699655136018697458, 2.50368180196714403235931239629, 3.18093684708761575735433762568, 3.62627962375111411256766350452, 4.20950183504054508342572519817, 4.45026094027893276220404624101, 4.51024131722509948981894231522, 5.28179898313075499340992032338, 5.47522858693247873179393504889, 6.28367676954311908126695486635, 6.50001199665982583334120459203, 6.77760590509000362233881941072, 7.27062063980197453505704949253, 7.74965789236081186774395995966, 7.76067301561830944874172980745, 8.260881621114192979309572123975, 8.846627195615937834471654949274
