| L(s) = 1 | + 12·4-s − 20·11-s + 80·16-s + 226·19-s − 440·29-s − 378·31-s + 260·41-s − 240·44-s + 686·49-s − 1.12e3·59-s + 458·61-s + 192·64-s − 1.78e3·71-s + 2.71e3·76-s + 54·79-s − 1.50e3·89-s + 3.00e3·101-s + 1.21e3·109-s − 5.28e3·116-s − 2.36e3·121-s − 4.53e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 0.548·11-s + 5/4·16-s + 2.72·19-s − 2.81·29-s − 2.19·31-s + 0.990·41-s − 0.822·44-s + 2·49-s − 2.47·59-s + 0.961·61-s + 3/8·64-s − 2.97·71-s + 4.09·76-s + 0.0769·79-s − 1.78·89-s + 2.95·101-s + 1.06·109-s − 4.22·116-s − 1.77·121-s − 3.28·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.530193284\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.530193284\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 p^{2} T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 10 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 2006 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9777 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 113 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 17773 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 220 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 189 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 72406 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 130 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 158914 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 182046 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 100407 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 560 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 229 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 39026 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 890 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 14066 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 27 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 959533 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 750 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 365054 T^{2} + p^{6} T^{4} \) |
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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
−10.52143236766713597658134041711, −9.907713477184204484605221415143, −9.400067010726692039243052749600, −9.150497669149293913884206246299, −8.713232042542003427866534240252, −7.69260477326695484500685855852, −7.50371092527234035558663178146, −7.40826733967362039155397611684, −7.10659131205646835757213527511, −6.08037986520764077242787618494, −5.95978832229188380569612746492, −5.28272755108175253935489059107, −5.25965375179881193956801173645, −4.09640366245980505143558125483, −3.66543007727849610503443400731, −3.01020664881058470256761280599, −2.69942477905901860821555837269, −1.76008436619441078991894837617, −1.56639156665414207446638037440, −0.49684717324468408313448198512,
0.49684717324468408313448198512, 1.56639156665414207446638037440, 1.76008436619441078991894837617, 2.69942477905901860821555837269, 3.01020664881058470256761280599, 3.66543007727849610503443400731, 4.09640366245980505143558125483, 5.25965375179881193956801173645, 5.28272755108175253935489059107, 5.95978832229188380569612746492, 6.08037986520764077242787618494, 7.10659131205646835757213527511, 7.40826733967362039155397611684, 7.50371092527234035558663178146, 7.69260477326695484500685855852, 8.713232042542003427866534240252, 9.150497669149293913884206246299, 9.400067010726692039243052749600, 9.907713477184204484605221415143, 10.52143236766713597658134041711
