| L(s) = 1 | − 4-s + 95·9-s − 44·11-s + 81·16-s + 6·19-s + 442·29-s + 282·31-s − 95·36-s − 288·41-s + 44·44-s + 371·49-s + 172·59-s − 310·61-s − 225·64-s + 3.17e3·71-s − 6·76-s + 2.58e3·79-s + 5.34e3·81-s + 3.55e3·89-s − 4.18e3·99-s − 1.85e3·101-s + 1.64e3·109-s − 442·116-s + 1.21e3·121-s − 282·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/8·4-s + 3.51·9-s − 1.20·11-s + 1.26·16-s + 0.0724·19-s + 2.83·29-s + 1.63·31-s − 0.439·36-s − 1.09·41-s + 0.150·44-s + 1.08·49-s + 0.379·59-s − 0.650·61-s − 0.439·64-s + 5.30·71-s − 0.00905·76-s + 3.68·79-s + 7.33·81-s + 4.23·89-s − 4.24·99-s − 1.82·101-s + 1.44·109-s − 0.353·116-s + 0.909·121-s − 0.204·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(10.26051655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.26051655\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + p T )^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 + T^{2} - 5 p^{4} T^{4} + p^{6} T^{6} + p^{12} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 - 95 T^{2} + 3676 T^{4} - 95 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 53 p T^{2} + 54552 T^{4} - 53 p^{7} T^{6} + p^{12} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 6688 T^{2} + 19773454 T^{4} - 6688 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 5795 T^{2} + 28665316 T^{4} - 5795 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 3 T + 5114 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 2060 p T^{2} + 857131206 T^{4} - 2060 p^{7} T^{6} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 221 T + 39564 T^{2} - 221 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 141 T + 6374 T^{2} - 141 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 30655 T^{2} + 2910831256 T^{4} - 30655 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 144 T + 39598 T^{2} + 144 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 109372 T^{2} + 15260943894 T^{4} - 109372 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 329140 T^{2} + 21975462 p^{2} T^{4} - 329140 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 345895 T^{2} + 72577931016 T^{4} - 345895 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 86 T + 306510 T^{2} - 86 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 155 T + 399784 T^{2} + 155 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 976424 T^{2} + 412501678782 T^{4} - 976424 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 1587 T + 1329650 T^{2} - 1587 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 1525024 T^{2} + 884022947422 T^{4} - 1525024 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 1294 T + 1349454 T^{2} - 1294 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 1511816 T^{2} + 1128809299902 T^{4} - 1511816 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 1777 T + 2191512 T^{2} - 1777 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 1842928 T^{2} + 2417317638046 T^{4} - 1842928 p^{6} T^{6} + p^{12} T^{8} \) |
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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
−8.027931237161871424392257441560, −7.980775333456928087718060525912, −7.73216574883631815875090944214, −7.56588585431064936082739177274, −6.98980004613168497416407331010, −6.85581944849055318227905035804, −6.66123507156817981216777913513, −6.61284201340044846671770821204, −6.08810398602743430818688444520, −5.88275957735464605953974101622, −5.34444877455118688922572436657, −4.96016855131287772733825271484, −4.87008820175990593003947564022, −4.60183254571979644935984580149, −4.56920389883350203371250224072, −3.80884362878297945976619025775, −3.73225685174482483524338269278, −3.46382616359390149130769786299, −3.00890095887306138394614572558, −2.41681227933541063152020160258, −2.09739132238555092272592116348, −1.87125566795404075754337128300, −1.00029397773553886878634320796, −0.832830105290406522855055924617, −0.818520054576956611350462280492,
0.818520054576956611350462280492, 0.832830105290406522855055924617, 1.00029397773553886878634320796, 1.87125566795404075754337128300, 2.09739132238555092272592116348, 2.41681227933541063152020160258, 3.00890095887306138394614572558, 3.46382616359390149130769786299, 3.73225685174482483524338269278, 3.80884362878297945976619025775, 4.56920389883350203371250224072, 4.60183254571979644935984580149, 4.87008820175990593003947564022, 4.96016855131287772733825271484, 5.34444877455118688922572436657, 5.88275957735464605953974101622, 6.08810398602743430818688444520, 6.61284201340044846671770821204, 6.66123507156817981216777913513, 6.85581944849055318227905035804, 6.98980004613168497416407331010, 7.56588585431064936082739177274, 7.73216574883631815875090944214, 7.980775333456928087718060525912, 8.027931237161871424392257441560
