| L(s) = 1 | − 2·4-s − 16·9-s + 56·11-s − 29·16-s − 40·25-s + 56·29-s + 32·36-s − 112·44-s + 62·49-s + 92·64-s − 64·71-s − 304·79-s + 30·81-s − 896·99-s + 80·100-s + 296·109-s − 112·116-s + 1.47e3·121-s + 127-s + 131-s + 137-s + 139-s + 464·144-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.77·9-s + 5.09·11-s − 1.81·16-s − 8/5·25-s + 1.93·29-s + 8/9·36-s − 2.54·44-s + 1.26·49-s + 1.43·64-s − 0.901·71-s − 3.84·79-s + 0.370·81-s − 9.05·99-s + 4/5·100-s + 2.71·109-s − 0.965·116-s + 12.1·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 29/9·144-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500625 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9888582673\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9888582673\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2^2$ | \( 1 + 8 p T^{2} + p^{4} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 62 T^{2} + p^{4} T^{4} \) |
| good | 2 | $C_2^2$ | \( ( 1 + T^{2} + p^{4} T^{4} )^{2} \) |
| 3 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 + 328 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 538 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 88 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 914 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 482 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 2414 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 3002 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 1934 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 2458 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 2702 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 6872 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 3032 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 1426 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + 6658 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 76 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 8488 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 12602 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 13978 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−12.15407755763340745044458001232, −11.80332801492276377411359897480, −11.64517565664980042195073372917, −11.43611550519918476870411483327, −11.20220127128625722333414267273, −10.77547695345533703248418065152, −9.915596054724230591623728215155, −9.782032140975098959711852531290, −9.576194405005039155412961470630, −8.913510597786663923032495047755, −8.777850099212590674768029056044, −8.631793773264566698803235714984, −8.623927565483976770284001564022, −7.57595049557816489357280751582, −7.04964578408385943295610019383, −6.85632429740892726584274572918, −6.31938465392784357736936317211, −6.04654041436690225883313799724, −5.89416070913243298913157686627, −4.84772126649158131360788524174, −4.36093397936944276801177774641, −4.00872259635927720849430176214, −3.64430520376682616594345731742, −2.66111727330181023687594782936, −1.44973716394288436679186751702,
1.44973716394288436679186751702, 2.66111727330181023687594782936, 3.64430520376682616594345731742, 4.00872259635927720849430176214, 4.36093397936944276801177774641, 4.84772126649158131360788524174, 5.89416070913243298913157686627, 6.04654041436690225883313799724, 6.31938465392784357736936317211, 6.85632429740892726584274572918, 7.04964578408385943295610019383, 7.57595049557816489357280751582, 8.623927565483976770284001564022, 8.631793773264566698803235714984, 8.777850099212590674768029056044, 8.913510597786663923032495047755, 9.576194405005039155412961470630, 9.782032140975098959711852531290, 9.915596054724230591623728215155, 10.77547695345533703248418065152, 11.20220127128625722333414267273, 11.43611550519918476870411483327, 11.64517565664980042195073372917, 11.80332801492276377411359897480, 12.15407755763340745044458001232
