| L(s) = 1 | + 4·4-s + 6·9-s − 12·11-s + 4·16-s − 24·19-s + 12·29-s + 24·31-s + 24·36-s − 8·41-s − 48·44-s − 8·59-s − 16·64-s − 24·71-s − 96·76-s + 28·79-s + 17·81-s + 32·89-s − 72·99-s − 24·101-s − 20·109-s + 48·116-s + 62·121-s + 96·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 2·4-s + 2·9-s − 3.61·11-s + 16-s − 5.50·19-s + 2.22·29-s + 4.31·31-s + 4·36-s − 1.24·41-s − 7.23·44-s − 1.04·59-s − 2·64-s − 2.84·71-s − 11.0·76-s + 3.15·79-s + 17/9·81-s + 3.39·89-s − 7.23·99-s − 2.38·101-s − 1.91·109-s + 4.45·116-s + 5.63·121-s + 8.62·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.279014823\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.279014823\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 3 | $D_4\times C_2$ | \( 1 - 2 p T^{2} + 19 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 30 T^{2} + 491 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 30 T^{2} + 43 p T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 834 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 12 T + 80 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 104 T^{2} + 5154 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 68 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 134 T^{2} + 8259 T^{4} - 134 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 88 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 + 4 T + 104 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 114 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 200 T^{2} + 17826 T^{4} - 200 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_4$ | \( ( 1 + 12 T + 170 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 14 T + 135 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 97 | $D_4\times C_2$ | \( 1 - 190 T^{2} + 22011 T^{4} - 190 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.76089548256995444868030985787, −6.67053471270795524994261000518, −6.53326402621313878751977596704, −6.33145812703529196274661043619, −6.30625124040644672737329991808, −6.19261642029639959108665378144, −5.56780775405479700421169115324, −5.51870060720393095539463448609, −4.91330903903926685933971161275, −4.82824007972504767333833889388, −4.69419364329223309662316416713, −4.54104191858632295215648241150, −4.27587111079545816388374276266, −4.11419967751325030347537872713, −3.81586321905988346677616454748, −3.07013687358727218921082103343, −2.91641999633163643795672602249, −2.72720781824124058285194022912, −2.68907717788295849296329943438, −2.17246388166365842908076329541, −2.12974941038525629259318311368, −1.87015858748459871769252328118, −1.55086579142035652647913968986, −0.824315677891015449332366281833, −0.30563185975764321630357699312,
0.30563185975764321630357699312, 0.824315677891015449332366281833, 1.55086579142035652647913968986, 1.87015858748459871769252328118, 2.12974941038525629259318311368, 2.17246388166365842908076329541, 2.68907717788295849296329943438, 2.72720781824124058285194022912, 2.91641999633163643795672602249, 3.07013687358727218921082103343, 3.81586321905988346677616454748, 4.11419967751325030347537872713, 4.27587111079545816388374276266, 4.54104191858632295215648241150, 4.69419364329223309662316416713, 4.82824007972504767333833889388, 4.91330903903926685933971161275, 5.51870060720393095539463448609, 5.56780775405479700421169115324, 6.19261642029639959108665378144, 6.30625124040644672737329991808, 6.33145812703529196274661043619, 6.53326402621313878751977596704, 6.67053471270795524994261000518, 6.76089548256995444868030985787
