| L(s) = 1 | − 2·3-s − 3·9-s − 4·13-s + 6·17-s + 12·23-s + 25-s + 14·27-s + 8·39-s − 2·43-s + 5·49-s − 12·51-s + 12·53-s + 16·61-s − 24·69-s − 2·75-s − 20·79-s − 4·81-s − 24·101-s − 28·103-s − 24·107-s − 12·113-s + 12·117-s + 22·121-s + 127-s + 4·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 9-s − 1.10·13-s + 1.45·17-s + 2.50·23-s + 1/5·25-s + 2.69·27-s + 1.28·39-s − 0.304·43-s + 5/7·49-s − 1.68·51-s + 1.64·53-s + 2.04·61-s − 2.88·69-s − 0.230·75-s − 2.25·79-s − 4/9·81-s − 2.38·101-s − 2.75·103-s − 2.32·107-s − 1.12·113-s + 1.10·117-s + 2·121-s + 0.0887·127-s + 0.352·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 692224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 692224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9463924117\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9463924117\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 83 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−10.40241192010663952315508800322, −10.26910105015602960943519560787, −9.466143678611589554249671227289, −9.318593395356145573227803132274, −8.738224145988317911600520077007, −8.217127924989861779845212682752, −8.051353117869808102241222519667, −7.15718494244760599904834015578, −6.88250171794278313779707583700, −6.74834796005373286005676871638, −5.71624689035849455832192117030, −5.50680028387874027494653052354, −5.40938207705914894450418727742, −4.81760581476506118633535067223, −4.22341099024535908985055972602, −3.42640587572810905482895313427, −2.71159958466673162444979031650, −2.69618801577100776364986294736, −1.27829586900682559029887878410, −0.57049774952064086039627902823,
0.57049774952064086039627902823, 1.27829586900682559029887878410, 2.69618801577100776364986294736, 2.71159958466673162444979031650, 3.42640587572810905482895313427, 4.22341099024535908985055972602, 4.81760581476506118633535067223, 5.40938207705914894450418727742, 5.50680028387874027494653052354, 5.71624689035849455832192117030, 6.74834796005373286005676871638, 6.88250171794278313779707583700, 7.15718494244760599904834015578, 8.051353117869808102241222519667, 8.217127924989861779845212682752, 8.738224145988317911600520077007, 9.318593395356145573227803132274, 9.466143678611589554249671227289, 10.26910105015602960943519560787, 10.40241192010663952315508800322
