| L(s) = 1 | − 77·4-s − 496·11-s + 2.50e3·16-s − 2.92e3·19-s + 3.89e3·29-s + 5.34e3·31-s + 1.52e4·41-s + 3.81e4·44-s + 8.60e3·49-s − 1.80e3·59-s + 4.04e4·61-s − 7.46e3·64-s + 8.19e4·71-s + 2.25e5·76-s − 2.15e5·79-s + 2.07e5·89-s − 1.26e5·101-s + 1.66e5·109-s − 2.99e5·116-s − 2.81e5·121-s − 4.11e5·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 2.40·4-s − 1.23·11-s + 2.44·16-s − 1.86·19-s + 0.860·29-s + 0.998·31-s + 1.41·41-s + 2.97·44-s + 0.511·49-s − 0.0676·59-s + 1.39·61-s − 0.227·64-s + 1.92·71-s + 4.47·76-s − 3.87·79-s + 2.77·89-s − 1.23·101-s + 1.34·109-s − 2.06·116-s − 1.74·121-s − 2.40·124-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8718766203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8718766203\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 + 77 T^{2} + 857 p^{2} T^{4} + 77 p^{10} T^{6} + p^{20} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 8604 T^{2} + 255105958 T^{4} - 8604 p^{10} T^{6} + p^{20} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 248 T + 232774 T^{2} + 248 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 1049132 T^{2} + 530799637878 T^{4} - 1049132 p^{10} T^{6} + p^{20} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 2297980 T^{2} + 2639484118598 T^{4} - 2297980 p^{10} T^{6} + p^{20} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 1464 T + 1064278 T^{2} + 1464 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 20102876 T^{2} + 181447864024038 T^{4} - 20102876 p^{10} T^{6} + p^{20} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 1948 T + 41552158 T^{2} - 1948 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 2672 T + 55875902 T^{2} - 2672 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 103641164 T^{2} + 7187119786904598 T^{4} - 103641164 p^{10} T^{6} + p^{20} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 7628 T + 215999542 T^{2} - 7628 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 452059340 T^{2} + 94199181438819798 T^{4} - 452059340 p^{10} T^{6} + p^{20} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 419580220 T^{2} + 91040640330233798 T^{4} - 419580220 p^{10} T^{6} + p^{20} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 514108876 T^{2} + 307072471963038038 T^{4} - 514108876 p^{10} T^{6} + p^{20} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 904 T + 361967398 T^{2} + 904 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 20220 T + 1540010398 T^{2} - 20220 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 74361060 p T^{2} + 9823337089416780598 T^{4} - 74361060 p^{11} T^{6} + p^{20} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 40976 T + 2981176846 T^{2} - 40976 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 6468866396 T^{2} + 18924680788840352358 T^{4} - 6468866396 p^{10} T^{6} + p^{20} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 107600 T + 7602983198 T^{2} + 107600 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 4978861292 T^{2} + 12452255953341624438 T^{4} - 4978861292 p^{10} T^{6} + p^{20} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 103764 T + 8272750678 T^{2} - 103764 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 2191776380 T^{2} + \)\(13\!\cdots\!98\)\( T^{4} - 2191776380 p^{10} T^{6} + p^{20} T^{8} \) |
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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
−8.264416961267077949114097842862, −7.79363142378744347638875416797, −7.53903937983875800292849262461, −7.50725484904993475888608788816, −6.90107297985490860001087204791, −6.49218249629099956169516261604, −6.44794396280500091935535695464, −6.19189306713891251602660036750, −5.57941751181165209779584136223, −5.37092108465059593398383724446, −5.31255713392338116656910302826, −4.86547262246443602883011514205, −4.55424379575721036143089605842, −4.34316470039725484411152441260, −4.14296206727227634684201706648, −3.92769050938431475707858351299, −3.45110890286356187402206795200, −2.97794681175137853399664952931, −2.67461765728889621935985648969, −2.15962282504506818771271249774, −2.14292893221631873131900349292, −1.19835998230298785337734211958, −0.953780874301898028411981747505, −0.39337516684535512165851021729, −0.29272600778437415796873056033,
0.29272600778437415796873056033, 0.39337516684535512165851021729, 0.953780874301898028411981747505, 1.19835998230298785337734211958, 2.14292893221631873131900349292, 2.15962282504506818771271249774, 2.67461765728889621935985648969, 2.97794681175137853399664952931, 3.45110890286356187402206795200, 3.92769050938431475707858351299, 4.14296206727227634684201706648, 4.34316470039725484411152441260, 4.55424379575721036143089605842, 4.86547262246443602883011514205, 5.31255713392338116656910302826, 5.37092108465059593398383724446, 5.57941751181165209779584136223, 6.19189306713891251602660036750, 6.44794396280500091935535695464, 6.49218249629099956169516261604, 6.90107297985490860001087204791, 7.50725484904993475888608788816, 7.53903937983875800292849262461, 7.79363142378744347638875416797, 8.264416961267077949114097842862
