| L(s) = 1 | − 16·13-s + 72·17-s + 10·25-s + 72·29-s − 80·37-s + 48·41-s + 112·49-s − 48·53-s + 256·61-s − 152·73-s + 18·81-s − 24·89-s − 104·97-s + 120·101-s − 64·109-s − 456·113-s + 340·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 156·169-s + ⋯ |
| L(s) = 1 | − 1.23·13-s + 4.23·17-s + 2/5·25-s + 2.48·29-s − 2.16·37-s + 1.17·41-s + 16/7·49-s − 0.905·53-s + 4.19·61-s − 2.08·73-s + 2/9·81-s − 0.269·89-s − 1.07·97-s + 1.18·101-s − 0.587·109-s − 4.03·113-s + 2.80·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 0.923·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.339850331\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.339850331\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 3 | $C_2^3$ | \( 1 - 2 p^{2} T^{4} + p^{8} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 16 p T^{2} + 6318 T^{4} - 16 p^{5} T^{6} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 340 T^{2} + 55302 T^{4} - 340 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 8 T + 174 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 18 T + p^{2} T^{2} )^{4} \) |
| 19 | $D_4\times C_2$ | \( 1 - 100 T^{2} - 151578 T^{4} - 100 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1360 T^{2} + 1000302 T^{4} - 1360 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 36 T + 1286 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 820 T^{2} + 2022 p^{2} T^{4} - 820 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 40 T + 2958 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 24 T + 46 p T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 5920 T^{2} + 15590382 T^{4} - 5920 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 592 T^{2} - 6245202 T^{4} - 592 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 24 T + 4142 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 11620 T^{2} + 57253542 T^{4} - 11620 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 128 T + 11358 T^{2} - 128 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 15520 T^{2} + 100505262 T^{4} - 15520 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 10228 T^{2} + 56168358 T^{4} - 10228 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 76 T + 9222 T^{2} + 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 10564 T^{2} + 76999686 T^{4} - 10564 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 224 p T^{2} + 165454638 T^{4} - 224 p^{5} T^{6} + p^{8} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 12 T + 12998 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 52 T + 1494 T^{2} + 52 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−8.256658108359328561975670986852, −7.87207188036710691323161784764, −7.79876040426896312231135348081, −7.42957995482404087322484739708, −7.20762674324186764795211622504, −7.03837170044908272976522040656, −6.82666060148061990724109302561, −6.38330528048468494767023558542, −6.12562135076906884646180380720, −5.76122042257327485584309035972, −5.45996319226312749054783557262, −5.34401460526845711408992377623, −5.10480180037302289148853039960, −4.94603581459413221620684553329, −4.22599017357692908189155399042, −4.19658452157222377852499453232, −3.78769037911237320577150383285, −3.31139074840849332981094376640, −3.14330157745783462108446927244, −2.78595069915151970540988332848, −2.50509635212001684715592164834, −1.96389670856562315505537350462, −1.27398856153856427168139671304, −1.02442044487613205647319190127, −0.57159260950025737000280026094,
0.57159260950025737000280026094, 1.02442044487613205647319190127, 1.27398856153856427168139671304, 1.96389670856562315505537350462, 2.50509635212001684715592164834, 2.78595069915151970540988332848, 3.14330157745783462108446927244, 3.31139074840849332981094376640, 3.78769037911237320577150383285, 4.19658452157222377852499453232, 4.22599017357692908189155399042, 4.94603581459413221620684553329, 5.10480180037302289148853039960, 5.34401460526845711408992377623, 5.45996319226312749054783557262, 5.76122042257327485584309035972, 6.12562135076906884646180380720, 6.38330528048468494767023558542, 6.82666060148061990724109302561, 7.03837170044908272976522040656, 7.20762674324186764795211622504, 7.42957995482404087322484739708, 7.79876040426896312231135348081, 7.87207188036710691323161784764, 8.256658108359328561975670986852
