| L(s) = 1 | − 2.58e4·9-s − 2.34e5·17-s + 1.42e6·25-s + 1.72e7·41-s + 1.30e6·49-s − 5.09e7·73-s + 4.15e8·81-s + 6.72e7·89-s + 4.83e8·97-s − 1.85e8·113-s + 2.14e8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 6.07e9·153-s + 157-s + 163-s + 167-s + 2.53e9·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 3.94·9-s − 2.81·17-s + 3.65·25-s + 6.12·41-s + 0.226·49-s − 1.79·73-s + 9.65·81-s + 1.07·89-s + 5.46·97-s − 1.13·113-s + 1.00·121-s + 11.0·153-s + 3.10·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32}\right)^{s/2} \, \Gamma_{\C}(s+4)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(3.734836149\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.734836149\) |
| \(L(5)\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 + 4310 p T^{2} + p^{16} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 - 714686 T^{2} + p^{16} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 13346 p^{2} T^{2} + p^{16} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 107392510 T^{2} + p^{16} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 1265198398 T^{2} + p^{16} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 58686 T + p^{8} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 10688638850 T^{2} + p^{16} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 83658891650 T^{2} + p^{16} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 291229042238 T^{2} + p^{16} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 600755547650 T^{2} + p^{16} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 529545975742 T^{2} + p^{16} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 4324158 T + p^{8} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 19226964507650 T^{2} + p^{16} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 4696760080126 T^{2} + p^{16} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 123098054289086 T^{2} + p^{16} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 293546462999810 T^{2} + p^{16} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 312606002568766 T^{2} + p^{16} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 508514414548610 T^{2} + p^{16} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 338146626840194 T^{2} + p^{16} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 12735874 T + p^{8} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 2994695631333122 T^{2} + p^{16} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 2396699180600446 T^{2} + p^{16} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 16802814 T + p^{8} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 120994882 T + p^{8} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.31520561963748664228091920895, −6.88835055450689182750503616505, −6.67156915004048070960772819357, −6.41232000822852260821658688377, −6.26696071745131936154074613150, −5.94028703473787460013989971233, −5.69208014401082100221203690044, −5.45879787529922037385831042392, −5.35994742699107298768280574857, −4.67589519817254481119298059662, −4.54133108387270780665879203104, −4.46219615205392963984933219713, −4.22463024578391707643913529146, −3.33781837721421658919409430267, −3.27936442285899302716984317371, −3.17846122685066989568505915701, −2.68511401255021603826657176202, −2.36918223673450665015796846782, −2.34798150102669242170653973324, −2.24601577978691111351393498416, −1.51891583309297368156559084108, −0.78419256258235150550322096329, −0.67827759917944040345653460861, −0.64657073557829663703219273284, −0.28470546811848353113541074893,
0.28470546811848353113541074893, 0.64657073557829663703219273284, 0.67827759917944040345653460861, 0.78419256258235150550322096329, 1.51891583309297368156559084108, 2.24601577978691111351393498416, 2.34798150102669242170653973324, 2.36918223673450665015796846782, 2.68511401255021603826657176202, 3.17846122685066989568505915701, 3.27936442285899302716984317371, 3.33781837721421658919409430267, 4.22463024578391707643913529146, 4.46219615205392963984933219713, 4.54133108387270780665879203104, 4.67589519817254481119298059662, 5.35994742699107298768280574857, 5.45879787529922037385831042392, 5.69208014401082100221203690044, 5.94028703473787460013989971233, 6.26696071745131936154074613150, 6.41232000822852260821658688377, 6.67156915004048070960772819357, 6.88835055450689182750503616505, 7.31520561963748664228091920895
