| L(s) = 1 | − 2·2-s + 2·4-s − 8·7-s − 4·8-s + 6·9-s − 12·13-s + 16·14-s + 8·16-s − 12·17-s − 12·18-s − 24·19-s − 8·23-s + 24·26-s − 16·28-s − 4·29-s − 24·31-s − 8·32-s + 24·34-s + 12·36-s + 48·38-s + 8·43-s + 16·46-s + 34·49-s − 24·52-s + 32·56-s + 8·58-s + 48·62-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 3.02·7-s − 1.41·8-s + 2·9-s − 3.32·13-s + 4.27·14-s + 2·16-s − 2.91·17-s − 2.82·18-s − 5.50·19-s − 1.66·23-s + 4.70·26-s − 3.02·28-s − 0.742·29-s − 4.31·31-s − 1.41·32-s + 4.11·34-s + 2·36-s + 7.78·38-s + 1.21·43-s + 2.35·46-s + 34/7·49-s − 3.32·52-s + 4.27·56-s + 1.05·58-s + 6.09·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 30 T^{2} + 419 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 + 6 T + 31 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 23 | $D_{4}$ | \( ( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 2 T + 11 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1686 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 91 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 148 T^{2} + 11190 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 228 T^{2} + 22310 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 94 T^{2} + 3891 T^{4} - 94 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 6966 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 31014 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 6 T + 95 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−8.032776293192002675944300816606, −7.56120126439705187181673483387, −7.47956817201514340032163902001, −7.16874214232023280176066163320, −7.06959897838279528725988942295, −6.78787963755710182953511468169, −6.68459994049540358915687053246, −6.68354836189646386812100605066, −6.20995323802689256026159277884, −5.84482706618288067489072645249, −5.84047545893985975467415757098, −5.78720637507001882686884694114, −5.05001172950635862879963783820, −4.67885885569612435376733762452, −4.52267099842250700225821745564, −4.06706948602700931474719011142, −4.04404603261967170687404738390, −4.02684444136377188015058673705, −3.60443275592825119503335231720, −2.99605312371938134764683502880, −2.60952557767775544407385728402, −2.48450189962822176728262392490, −2.06832379625223456238426573275, −1.99889863511187345292225532866, −1.76987501835780290750468231858, 0, 0, 0, 0,
1.76987501835780290750468231858, 1.99889863511187345292225532866, 2.06832379625223456238426573275, 2.48450189962822176728262392490, 2.60952557767775544407385728402, 2.99605312371938134764683502880, 3.60443275592825119503335231720, 4.02684444136377188015058673705, 4.04404603261967170687404738390, 4.06706948602700931474719011142, 4.52267099842250700225821745564, 4.67885885569612435376733762452, 5.05001172950635862879963783820, 5.78720637507001882686884694114, 5.84047545893985975467415757098, 5.84482706618288067489072645249, 6.20995323802689256026159277884, 6.68354836189646386812100605066, 6.68459994049540358915687053246, 6.78787963755710182953511468169, 7.06959897838279528725988942295, 7.16874214232023280176066163320, 7.47956817201514340032163902001, 7.56120126439705187181673483387, 8.032776293192002675944300816606
