| L(s) = 1 | − 176·5-s − 4.09e3·16-s − 1.95e4·17-s − 2.86e4·25-s + 6.37e4·29-s − 1.69e5·37-s + 1.69e5·41-s + 5.33e5·53-s + 7.76e5·61-s + 8.55e5·73-s + 7.20e5·80-s + 3.44e6·85-s − 2.15e6·97-s − 5.09e6·109-s − 5.31e6·113-s + 9.14e6·125-s + 127-s + 131-s + 137-s + 139-s − 1.12e7·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.34e7·169-s + ⋯ |
| L(s) = 1 | − 1.40·5-s − 16-s − 3.97·17-s − 1.83·25-s + 2.61·29-s − 3.34·37-s + 2.46·41-s + 3.58·53-s + 3.42·61-s + 2.19·73-s + 1.40·80-s + 5.60·85-s − 2.36·97-s − 3.93·109-s − 3.68·113-s + 4.68·125-s − 3.68·145-s + 2.79·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21381376 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21381376 ^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.7111485603\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7111485603\) |
| \(L(4)\) |
|
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 | $C_2^2$ | \( 1 + p^{12} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 9776 T + p^{6} T^{2} )^{2} \) |
| good | 3 | $C_4\times C_2$ | \( 1 + p^{24} T^{8} \) |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + 88 T + p^{6} T^{2} )^{2}( 1 + 20592 T^{2} + p^{12} T^{4} ) \) |
| 7 | $C_4\times C_2$ | \( 1 + p^{24} T^{8} \) |
| 11 | $C_4\times C_2$ | \( 1 + p^{24} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 6739920 T^{2} + p^{12} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + p^{12} T^{4} )^{2} \) |
| 23 | $C_4\times C_2$ | \( 1 + p^{24} T^{8} \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 - 31878 T + p^{6} T^{2} )^{2}( 1 + 1176935760 T^{2} + p^{12} T^{4} ) \) |
| 31 | $C_4\times C_2$ | \( 1 + p^{24} T^{8} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 + 84744 T + p^{6} T^{2} )^{2}( 1 - 4704139440 T^{2} + p^{12} T^{4} ) \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 - 84942 T + p^{6} T^{2} )^{2}( 1 + 9221303520 T^{2} + p^{12} T^{4} ) \) |
| 43 | $C_2^2$ | \( ( 1 + p^{12} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 - 296296 T + p^{6} T^{2} )^{2}( 1 + 29430 T + p^{6} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + p^{12} T^{4} )^{2} \) |
| 61 | $C_2$$\times$$C_2^2$ | \( ( 1 - 388440 T + p^{6} T^{2} )^{2}( 1 - 91259316720 T^{2} + p^{12} T^{4} ) \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{4}( 1 + p^{3} T )^{4} \) |
| 71 | $C_4\times C_2$ | \( 1 + p^{24} T^{8} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 - 427570 T + p^{6} T^{2} )^{2}( 1 - 277927341120 T^{2} + p^{12} T^{4} ) \) |
| 79 | $C_4\times C_2$ | \( 1 + p^{24} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 + p^{12} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 405304511040 T^{2} + p^{12} T^{4} )^{2} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + 1078704 T + p^{6} T^{2} )^{2}( 1 - 1588402427040 T^{2} + p^{12} T^{4} ) \) |
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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
−9.829743419736729059623152384088, −9.174143694855644162406435758532, −9.111708263142811497238340700381, −8.565147314980665511831420944469, −8.547543012205895088868676352485, −8.139491332182401011114387612294, −8.057049975438256340893579204440, −7.23557576589360377201591479752, −7.15504919177728068200539076712, −6.77649820609899098106385817495, −6.59597844620837462608312020571, −6.39780312747177124741228010181, −5.50946720396401272652894856540, −5.43565653923283302012799142607, −4.88671097482351154028127269237, −4.29864650357103124109063560164, −4.21887011328680775087928557418, −3.87003014821807300108692738236, −3.69990370845696999264256209721, −2.46834035658699803440546193013, −2.46157135895829409362993080141, −2.26057178729209085717751007615, −1.37406457522210978181904261946, −0.51117444832757390027373757673, −0.26122232670541022795825563231,
0.26122232670541022795825563231, 0.51117444832757390027373757673, 1.37406457522210978181904261946, 2.26057178729209085717751007615, 2.46157135895829409362993080141, 2.46834035658699803440546193013, 3.69990370845696999264256209721, 3.87003014821807300108692738236, 4.21887011328680775087928557418, 4.29864650357103124109063560164, 4.88671097482351154028127269237, 5.43565653923283302012799142607, 5.50946720396401272652894856540, 6.39780312747177124741228010181, 6.59597844620837462608312020571, 6.77649820609899098106385817495, 7.15504919177728068200539076712, 7.23557576589360377201591479752, 8.057049975438256340893579204440, 8.139491332182401011114387612294, 8.547543012205895088868676352485, 8.565147314980665511831420944469, 9.111708263142811497238340700381, 9.174143694855644162406435758532, 9.829743419736729059623152384088
