L(s) = 1 | + 3-s + 5-s − 3·7-s + 3·9-s − 2·11-s − 5·13-s + 15-s − 10·17-s + 14·19-s − 3·21-s − 5·23-s + 2·25-s + 8·27-s + 3·29-s − 7·31-s − 2·33-s − 3·35-s + 12·37-s − 5·39-s + 12·41-s − 8·43-s + 3·45-s − 3·47-s + 8·49-s − 10·51-s − 20·53-s − 2·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.13·7-s + 9-s − 0.603·11-s − 1.38·13-s + 0.258·15-s − 2.42·17-s + 3.21·19-s − 0.654·21-s − 1.04·23-s + 2/5·25-s + 1.53·27-s + 0.557·29-s − 1.25·31-s − 0.348·33-s − 0.507·35-s + 1.97·37-s − 0.800·39-s + 1.87·41-s − 1.21·43-s + 0.447·45-s − 0.437·47-s + 8/7·49-s − 1.40·51-s − 2.74·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7977493164\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7977493164\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
good | 5 | $D_4\times C_2$ | \( 1 - T - T^{2} + 8 T^{3} - 26 T^{4} + 8 p T^{5} - p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 3 T + T^{2} - 18 T^{3} - 48 T^{4} - 18 p T^{5} + p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 5 T + T^{2} - 10 T^{3} + 82 T^{4} - 10 p T^{5} + p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 5 T + 32 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 7 T + 42 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 5 T - 19 T^{2} - 10 T^{3} + 832 T^{4} - 10 p T^{5} - 19 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 3 T - 43 T^{2} + 18 T^{3} + 1602 T^{4} + 18 p T^{5} - 43 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 7 T - 17 T^{2} + 28 T^{3} + 1876 T^{4} + 28 p T^{5} - 17 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 12 T + 59 T^{2} - 36 T^{3} - 360 T^{4} - 36 p T^{5} + 59 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 8 T - 5 T^{2} - 136 T^{3} + 160 T^{4} - 136 p T^{5} - 5 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 3 T - 79 T^{2} - 18 T^{3} + 5112 T^{4} - 18 p T^{5} - 79 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 10 T + 98 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 7 T - 10 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - T - 113 T^{2} + 8 T^{3} + 9214 T^{4} + 8 p T^{5} - 113 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 4 T - 89 T^{2} - 116 T^{3} + 5464 T^{4} - 116 p T^{5} - 89 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 73 | $D_{4}$ | \( ( 1 + 7 T + 84 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 7 T - 113 T^{2} - 28 T^{3} + 16132 T^{4} - 28 p T^{5} - 113 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 25 T + 311 T^{2} + 3700 T^{3} + 39832 T^{4} + 3700 p T^{5} + 311 p^{2} T^{6} + 25 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 97 | $D_4\times C_2$ | \( 1 - 8 T - 113 T^{2} + 136 T^{3} + 15712 T^{4} + 136 p T^{5} - 113 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−10.97484239703290360874038635021, −10.33316642614616915853722407192, −10.31658830517586373712031971696, −9.849785781643367104460999903642, −9.712103960320424840175102590290, −9.585166706487300286107305189198, −9.009342894345427883741841716162, −8.919556668285640819836691952917, −8.787589947031332386701770994099, −7.909028627599230400162437591922, −7.59561004699652374725667452262, −7.52659070916985522092147891192, −7.37605072270638361810659625308, −6.78796326320592240347188043600, −6.30787816442194039886959575366, −6.25523529252752935890943165888, −5.78935191560722170999123948828, −5.06172920739721406035955948384, −4.93524655292304400119499297670, −4.40959661098213397712541338428, −4.17387174021514465059944413867, −3.11732750306004008507032683652, −3.08366219427620883466983368697, −2.57422165224831863614353373906, −1.73332716400736499919266292061,
1.73332716400736499919266292061, 2.57422165224831863614353373906, 3.08366219427620883466983368697, 3.11732750306004008507032683652, 4.17387174021514465059944413867, 4.40959661098213397712541338428, 4.93524655292304400119499297670, 5.06172920739721406035955948384, 5.78935191560722170999123948828, 6.25523529252752935890943165888, 6.30787816442194039886959575366, 6.78796326320592240347188043600, 7.37605072270638361810659625308, 7.52659070916985522092147891192, 7.59561004699652374725667452262, 7.909028627599230400162437591922, 8.787589947031332386701770994099, 8.919556668285640819836691952917, 9.009342894345427883741841716162, 9.585166706487300286107305189198, 9.712103960320424840175102590290, 9.849785781643367104460999903642, 10.31658830517586373712031971696, 10.33316642614616915853722407192, 10.97484239703290360874038635021