| L(s) = 1 | + 4·5-s − 8·11-s − 8·19-s + 8·25-s − 8·29-s + 8·31-s − 24·41-s + 20·49-s − 32·55-s − 8·59-s + 16·61-s − 8·71-s − 40·79-s − 24·89-s − 32·95-s − 40·101-s − 24·109-s + 8·121-s + 20·125-s + 127-s + 131-s + 137-s + 139-s − 32·145-s + 149-s + 151-s + 32·155-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 2.41·11-s − 1.83·19-s + 8/5·25-s − 1.48·29-s + 1.43·31-s − 3.74·41-s + 20/7·49-s − 4.31·55-s − 1.04·59-s + 2.04·61-s − 0.949·71-s − 4.50·79-s − 2.54·89-s − 3.28·95-s − 3.98·101-s − 2.29·109-s + 8/11·121-s + 1.78·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.65·145-s + 0.0819·149-s + 0.0813·151-s + 2.57·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3324772693\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3324772693\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 31 | $D_{4}$ | \( ( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 28 T^{2} - 522 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 12 T + 112 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 28 T^{2} + 2358 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 72 T^{2} + 4850 T^{4} - 72 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 156 T^{2} + 11318 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 4 T + 116 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 67 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 8598 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 4 T - 4 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 11238 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 83 | $D_4\times C_2$ | \( 1 - 248 T^{2} + 28290 T^{4} - 248 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 12 T + 160 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 158 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.28745792136328968471745054408, −6.24589704255022765035580789627, −5.87050275569049904232350478960, −5.63915955444519868142609573609, −5.48227113801218945643379468978, −5.37452976810041212452779027230, −5.37031009839671514420604166801, −5.07820959003222613731120105096, −4.65875489641995744274581863535, −4.61066552670091387209788828429, −4.28667270182059118852814008695, −3.93014523359381580524068960982, −3.86947874109693927900598170548, −3.76875834713987468846261772095, −3.05945200012905405283757465256, −2.85120293214824534801667493788, −2.73619529734347891320244880718, −2.56957424228578374966604196568, −2.54792253266930659633664207205, −1.95738092571347006588021651426, −1.63841629496502772080620280578, −1.62928941580069326528650220681, −1.38356181668985334176373164537, −0.58922800656703909911703653953, −0.10616266961978173243114885971,
0.10616266961978173243114885971, 0.58922800656703909911703653953, 1.38356181668985334176373164537, 1.62928941580069326528650220681, 1.63841629496502772080620280578, 1.95738092571347006588021651426, 2.54792253266930659633664207205, 2.56957424228578374966604196568, 2.73619529734347891320244880718, 2.85120293214824534801667493788, 3.05945200012905405283757465256, 3.76875834713987468846261772095, 3.86947874109693927900598170548, 3.93014523359381580524068960982, 4.28667270182059118852814008695, 4.61066552670091387209788828429, 4.65875489641995744274581863535, 5.07820959003222613731120105096, 5.37031009839671514420604166801, 5.37452976810041212452779027230, 5.48227113801218945643379468978, 5.63915955444519868142609573609, 5.87050275569049904232350478960, 6.24589704255022765035580789627, 6.28745792136328968471745054408
