| L(s) = 1 | + 23·4-s + 19·9-s + 160·11-s + 273·16-s + 168·19-s + 88·29-s − 172·31-s + 437·36-s − 460·41-s + 3.68e3·44-s + 303·49-s + 736·59-s − 2.11e3·61-s + 1.86e3·64-s − 262·71-s + 3.86e3·76-s + 2.01e3·79-s − 231·81-s + 1.44e3·89-s + 3.04e3·99-s + 2.51e3·101-s − 3.04e3·109-s + 2.02e3·116-s + 1.19e4·121-s − 3.95e3·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 23/8·4-s + 0.703·9-s + 4.38·11-s + 4.26·16-s + 2.02·19-s + 0.563·29-s − 0.996·31-s + 2.02·36-s − 1.75·41-s + 12.6·44-s + 0.883·49-s + 1.62·59-s − 4.44·61-s + 3.63·64-s − 0.437·71-s + 5.83·76-s + 2.87·79-s − 0.316·81-s + 1.71·89-s + 3.08·99-s + 2.47·101-s − 2.67·109-s + 1.62·116-s + 8.94·121-s − 2.86·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(26.66689905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(26.66689905\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 23 T^{2} + p^{8} T^{4} - 23 p^{6} T^{6} + p^{12} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 - 19 T^{2} + 592 T^{4} - 19 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 303 T^{2} + 216596 T^{4} - 303 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 80 T + 3650 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 17015 T^{2} + 120209296 T^{4} - 17015 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 84 T + 11130 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 28236 T^{2} + 471879110 T^{4} - 28236 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 44 T + 10094 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 86 T + 56518 T^{2} + 86 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 180559 T^{2} + 13277199832 T^{4} - 180559 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 230 T + 149010 T^{2} + 230 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 103027 T^{2} + 8137821136 T^{4} - 103027 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 196231 T^{2} + 19410698292 T^{4} - 196231 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 580896 T^{2} + 128635503662 T^{4} - 580896 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 368 T + 379266 T^{2} - 368 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 1058 T + 580378 T^{2} + 1058 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 743236 T^{2} + 317958953014 T^{4} - 743236 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 131 T + 493328 T^{2} + 131 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 877500 T^{2} + 435430002278 T^{4} - 877500 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 1008 T + 1233294 T^{2} - 1008 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 88712 T^{2} + 116142406846 T^{4} - 88712 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 720 T + 899726 T^{2} - 720 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 1026556 T^{2} + 984875817094 T^{4} - 1026556 p^{6} T^{6} + p^{12} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.56961256098435173752296635982, −7.56839771430920734485331135983, −7.56356312724501988160681225706, −7.08741802148926598695932822741, −6.72145977667546305861993317213, −6.71758275121854846080532643478, −6.67185060694338714844913457519, −6.31768915258954674563889033416, −6.08596167276612648476747002950, −5.76390430970749247955223197594, −5.58354731821765739370576775454, −5.02856730472699485228796782606, −4.76779479718188982490420179830, −4.23451181946820607491017048254, −4.19353380197353844081808288892, −3.48692005240418521227189660971, −3.46048721376878680794274874190, −3.39846612707465757554114803862, −2.84102768664515225058200573575, −2.39168006480201574429037631894, −1.78253068736606955519741748194, −1.64359821011770259688267907668, −1.57866269879032537770918170837, −1.01438531445432514586042981298, −0.75372406408650053710416698130,
0.75372406408650053710416698130, 1.01438531445432514586042981298, 1.57866269879032537770918170837, 1.64359821011770259688267907668, 1.78253068736606955519741748194, 2.39168006480201574429037631894, 2.84102768664515225058200573575, 3.39846612707465757554114803862, 3.46048721376878680794274874190, 3.48692005240418521227189660971, 4.19353380197353844081808288892, 4.23451181946820607491017048254, 4.76779479718188982490420179830, 5.02856730472699485228796782606, 5.58354731821765739370576775454, 5.76390430970749247955223197594, 6.08596167276612648476747002950, 6.31768915258954674563889033416, 6.67185060694338714844913457519, 6.71758275121854846080532643478, 6.72145977667546305861993317213, 7.08741802148926598695932822741, 7.56356312724501988160681225706, 7.56839771430920734485331135983, 7.56961256098435173752296635982
