| L(s) = 1 | − 2·2-s − 2·3-s + 4-s − 6·5-s + 4·6-s + 2·7-s + 2·8-s + 9-s + 12·10-s − 11-s − 2·12-s − 2·13-s − 4·14-s + 12·15-s − 4·16-s + 10·17-s − 2·18-s − 3·19-s − 6·20-s − 4·21-s + 2·22-s + 10·23-s − 4·24-s + 11·25-s + 4·26-s + 2·27-s + 2·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 1/2·4-s − 2.68·5-s + 1.63·6-s + 0.755·7-s + 0.707·8-s + 1/3·9-s + 3.79·10-s − 0.301·11-s − 0.577·12-s − 0.554·13-s − 1.06·14-s + 3.09·15-s − 16-s + 2.42·17-s − 0.471·18-s − 0.688·19-s − 1.34·20-s − 0.872·21-s + 0.426·22-s + 2.08·23-s − 0.816·24-s + 11/5·25-s + 0.784·26-s + 0.384·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2122375847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2122375847\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 + T - 17 T^{2} - 4 T^{3} + 192 T^{4} - 4 p T^{5} - 17 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 5 T + 8 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 + 3 T + 7 T^{2} - 108 T^{3} - 528 T^{4} - 108 p T^{5} + 7 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 10 T + 2 p T^{2} - 80 T^{3} + 87 T^{4} - 80 p T^{5} + 2 p^{3} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 2 T + 46 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + T + 33 T^{2} - 106 T^{3} - 382 T^{4} - 106 p T^{5} + 33 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 21 T + 188 T^{2} - 21 p T^{3} + p^{2} T^{4} )( 1 + 21 T + 188 T^{2} + 21 p T^{3} + p^{2} T^{4} ) \) |
| 43 | $D_4\times C_2$ | \( 1 - 14 T + 78 T^{2} - 448 T^{3} + 3647 T^{4} - 448 p T^{5} + 78 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 3 T + 58 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 16 T + 153 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 18 T + 142 T^{2} + 1152 T^{3} + 10527 T^{4} + 1152 p T^{5} + 142 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 6 T - 90 T^{2} + 48 T^{3} + 8975 T^{4} + 48 p T^{5} - 90 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 4 T - 55 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 9 T + 162 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 7 T + 166 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 7 T - 103 T^{2} + 182 T^{3} + 10822 T^{4} + 182 p T^{5} - 103 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 8 T - 78 T^{2} - 416 T^{3} + 4547 T^{4} - 416 p T^{5} - 78 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.66105301476937064089661115836, −7.62946073692979952448935302284, −7.59052919865805356299634279700, −7.25937245960365133829550218255, −7.23336688417032258737000325406, −6.60013991705357099876328372533, −6.33848361340608420257078706634, −6.23718076521617461231642562774, −5.79539041791953508252543694692, −5.77507233372255184933654097995, −5.08134273399618757730208809428, −4.88948270922628189036482240660, −4.86851453958344796612352960406, −4.68374896571471523754025836180, −4.36404261140080671322979389177, −4.00787513167301698174705292594, −3.55009733553347817523035492660, −3.43418833114470502007861530368, −3.05429821191537438574459869072, −2.91628802504928215060906728393, −2.19526854342959724842261703529, −1.64597133344588705047313029374, −1.23128666803714783207566773468, −0.72578109763899831388137551844, −0.35327922923447364433279661753,
0.35327922923447364433279661753, 0.72578109763899831388137551844, 1.23128666803714783207566773468, 1.64597133344588705047313029374, 2.19526854342959724842261703529, 2.91628802504928215060906728393, 3.05429821191537438574459869072, 3.43418833114470502007861530368, 3.55009733553347817523035492660, 4.00787513167301698174705292594, 4.36404261140080671322979389177, 4.68374896571471523754025836180, 4.86851453958344796612352960406, 4.88948270922628189036482240660, 5.08134273399618757730208809428, 5.77507233372255184933654097995, 5.79539041791953508252543694692, 6.23718076521617461231642562774, 6.33848361340608420257078706634, 6.60013991705357099876328372533, 7.23336688417032258737000325406, 7.25937245960365133829550218255, 7.59052919865805356299634279700, 7.62946073692979952448935302284, 7.66105301476937064089661115836
