| L(s) = 1 | − 4·4-s + 12·13-s + 12·16-s − 60·19-s + 16·31-s − 112·37-s + 220·43-s + 14·49-s − 48·52-s − 88·61-s − 32·64-s + 152·67-s − 84·73-s + 240·76-s + 144·79-s − 152·97-s − 372·103-s − 208·109-s + 20·121-s − 64·124-s + 127-s + 131-s + 137-s + 139-s + 448·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 4-s + 0.923·13-s + 3/4·16-s − 3.15·19-s + 0.516·31-s − 3.02·37-s + 5.11·43-s + 2/7·49-s − 0.923·52-s − 1.44·61-s − 1/2·64-s + 2.26·67-s − 1.15·73-s + 3.15·76-s + 1.82·79-s − 1.56·97-s − 3.61·103-s − 1.90·109-s + 0.165·121-s − 0.516·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 3.02·148-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6700998616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6700998616\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 11 | $D_4\times C_2$ | \( 1 - 20 T^{2} - 12121 T^{4} - 20 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 6 T + 340 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 112 T^{2} + 40706 T^{4} + 112 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 30 T + 772 T^{2} + 30 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 932 T^{2} + 775655 T^{4} - 932 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2492 T^{2} + 2785631 T^{4} - 2492 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 8 T + 1490 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 56 T + 3347 T^{2} + 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 4420 T^{2} + 10453974 T^{4} - 4420 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 110 T + 6471 T^{2} - 110 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 1960 T^{2} + 7090962 T^{4} - 1960 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 3632 T^{2} + 7466210 T^{4} - 3632 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 3952 T^{2} + 19974498 T^{4} - 3952 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 44 T + 3894 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 76 T + 10079 T^{2} - 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 6260 T^{2} + 48182039 T^{4} - 6260 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 42 T + 2524 T^{2} + 42 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 72 T + 11251 T^{2} - 72 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 11900 T^{2} + 120439734 T^{4} + 11900 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 11048 T^{2} + 54493010 T^{4} - 11048 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 76 T + 11190 T^{2} + 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−5.86578497209397784161999206394, −5.74394495144564640567858523633, −5.47022536627967569675839710819, −5.45523192053522958080778090927, −5.33476634531618700331485238918, −4.84239483637009344806129886284, −4.71637617418054324899244148108, −4.34801401097290343505887220180, −4.20269232067732036654469824029, −4.13437138118980492919772741942, −4.09822006875675163116371019821, −3.88357322085971501597539839361, −3.45817668611110288295407223272, −3.24260827515520788576588292151, −2.99715324513485070114919215182, −2.72168230290695917757850399328, −2.58662034948321490228758807385, −2.12079892675623457817552710597, −1.92861799983495473741630638145, −1.86696774425579545093517683651, −1.45880321557351396362952903938, −1.05686760678656321616465879740, −0.69990429820394205306483059870, −0.59503913222093295444856622402, −0.11351536860141599536272940175,
0.11351536860141599536272940175, 0.59503913222093295444856622402, 0.69990429820394205306483059870, 1.05686760678656321616465879740, 1.45880321557351396362952903938, 1.86696774425579545093517683651, 1.92861799983495473741630638145, 2.12079892675623457817552710597, 2.58662034948321490228758807385, 2.72168230290695917757850399328, 2.99715324513485070114919215182, 3.24260827515520788576588292151, 3.45817668611110288295407223272, 3.88357322085971501597539839361, 4.09822006875675163116371019821, 4.13437138118980492919772741942, 4.20269232067732036654469824029, 4.34801401097290343505887220180, 4.71637617418054324899244148108, 4.84239483637009344806129886284, 5.33476634531618700331485238918, 5.45523192053522958080778090927, 5.47022536627967569675839710819, 5.74394495144564640567858523633, 5.86578497209397784161999206394
