| L(s) = 1 | − 616·7-s + 3.69e4·13-s − 2.99e5·19-s + 7.79e5·25-s − 9.33e5·31-s − 1.92e6·37-s + 4.13e6·43-s − 1.12e7·49-s − 7.53e6·61-s − 5.24e7·67-s + 1.41e6·73-s − 7.69e7·79-s − 2.27e7·91-s − 2.22e8·97-s − 7.76e7·103-s − 3.94e8·109-s + 3.41e8·121-s + 127-s + 131-s + 1.84e8·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 0.256·7-s + 1.29·13-s − 2.29·19-s + 1.99·25-s − 1.01·31-s − 1.02·37-s + 1.20·43-s − 1.95·49-s − 0.544·61-s − 2.60·67-s + 0.0499·73-s − 1.97·79-s − 0.331·91-s − 2.51·97-s − 0.689·103-s − 2.79·109-s + 1.59·121-s + 0.588·133-s − 0.746·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.8699881188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8699881188\) |
| \(L(5)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 779792 T^{2} + p^{16} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 44 p T + p^{8} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 341914274 T^{2} + p^{16} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 18464 T + p^{8} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 123213760 T^{2} + p^{16} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 149552 T + p^{8} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 116577214 p^{2} T^{2} + p^{16} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 657268473680 T^{2} + p^{16} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 466532 T + p^{8} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 964522 T + p^{8} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3199918720640 T^{2} + p^{16} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 2067160 T + p^{8} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 32170265477090 T^{2} + p^{16} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 35895671640272 T^{2} + p^{16} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 233606563353410 T^{2} + p^{16} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 3766390 T + p^{8} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 26223512 T + p^{8} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 692317131053470 T^{2} + p^{16} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 709136 T + p^{8} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 38465660 T + p^{8} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 855631324702754 T^{2} + p^{16} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 3842243579471744 T^{2} + p^{16} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 111270688 T + p^{8} T^{2} )^{2} \) |
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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
−11.87174022291283144153830503148, −10.96815268702034646721579467437, −10.88693158241756214751416685964, −10.53783247262767241503576627256, −9.794683654202605496758210867552, −8.919933603833023712062914177167, −8.886037891933587907887240653006, −8.309429543436409704063116831151, −7.65705600677337111566177680037, −6.75214182612438913492256372114, −6.64029266732036237996614184314, −5.91634759780228753072648699846, −5.35017552827329753571166505340, −4.42005991478776026307782241703, −4.14311692767731447907255424630, −3.23152302512044785059245317223, −2.74561638055565496121731226438, −1.73525546735351727202685203396, −1.30172055488149637648426209360, −0.23855558646078690303280244525,
0.23855558646078690303280244525, 1.30172055488149637648426209360, 1.73525546735351727202685203396, 2.74561638055565496121731226438, 3.23152302512044785059245317223, 4.14311692767731447907255424630, 4.42005991478776026307782241703, 5.35017552827329753571166505340, 5.91634759780228753072648699846, 6.64029266732036237996614184314, 6.75214182612438913492256372114, 7.65705600677337111566177680037, 8.309429543436409704063116831151, 8.886037891933587907887240653006, 8.919933603833023712062914177167, 9.794683654202605496758210867552, 10.53783247262767241503576627256, 10.88693158241756214751416685964, 10.96815268702034646721579467437, 11.87174022291283144153830503148
