| L(s) = 1 | − 16·7-s + 8·11-s + 16·17-s − 48·23-s + 124·31-s + 32·37-s + 112·41-s − 112·43-s − 160·47-s + 128·49-s + 208·53-s + 300·61-s + 144·67-s + 272·71-s + 224·73-s − 128·77-s − 9·81-s − 160·83-s − 320·97-s + 224·101-s + 96·103-s − 144·107-s − 320·113-s − 256·119-s − 252·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 2.28·7-s + 8/11·11-s + 0.941·17-s − 2.08·23-s + 4·31-s + 0.864·37-s + 2.73·41-s − 2.60·43-s − 3.40·47-s + 2.61·49-s + 3.92·53-s + 4.91·61-s + 2.14·67-s + 3.83·71-s + 3.06·73-s − 1.66·77-s − 1/9·81-s − 1.92·83-s − 3.29·97-s + 2.21·101-s + 0.932·103-s − 1.34·107-s − 2.83·113-s − 2.15·119-s − 2.08·121-s + 0.00787·127-s + 0.00763·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.552027772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.552027772\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 864 T^{3} + 5807 T^{4} + 864 p^{2} T^{5} + 128 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 4 T + 150 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^3$ | \( 1 + 39599 T^{4} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 3408 T^{3} + 84962 T^{4} - 3408 p^{2} T^{5} + 128 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 286 T^{2} + 277635 T^{4} + 286 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 48 T + 1152 T^{2} + 1680 p T^{3} + 2306 p^{2} T^{4} + 1680 p^{3} T^{5} + 1152 p^{4} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1244 T^{2} + 1124070 T^{4} - 1244 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 2 p T + 2787 T^{2} - 2 p^{3} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 32 T + 512 T^{2} - 46368 T^{3} + 4192802 T^{4} - 46368 p^{2} T^{5} + 512 p^{4} T^{6} - 32 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 56 T + 4050 T^{2} - 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 112 T + 6272 T^{2} + 366240 T^{3} + 19366559 T^{4} + 366240 p^{2} T^{5} + 6272 p^{4} T^{6} + 112 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 160 T + 12800 T^{2} + 817440 T^{3} + 43793762 T^{4} + 817440 p^{2} T^{5} + 12800 p^{4} T^{6} + 160 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 208 T + 21632 T^{2} - 1699152 T^{3} + 104735522 T^{4} - 1699152 p^{2} T^{5} + 21632 p^{4} T^{6} - 208 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 238 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 150 T + 12683 T^{2} - 150 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 144 T + 10368 T^{2} - 863712 T^{3} + 69674927 T^{4} - 863712 p^{2} T^{5} + 10368 p^{4} T^{6} - 144 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 136 T + 14610 T^{2} - 136 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 224 T + 25088 T^{2} - 2071776 T^{3} + 155721602 T^{4} - 2071776 p^{2} T^{5} + 25088 p^{4} T^{6} - 224 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 5692 T^{2} + 84617478 T^{4} - 5692 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 160 T + 12800 T^{2} + 1520160 T^{3} + 173715458 T^{4} + 1520160 p^{2} T^{5} + 12800 p^{4} T^{6} + 160 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 8284 T^{2} + 112535046 T^{4} + 8284 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 320 T + 51200 T^{2} + 6760320 T^{3} + 755327663 T^{4} + 6760320 p^{2} T^{5} + 51200 p^{4} T^{6} + 320 p^{6} T^{7} + p^{8} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.67280993970042838402645348450, −6.64904589432122808647540687937, −6.35482594459534226370232423437, −6.24274651150057130257037224079, −6.03119454288081281747690953739, −5.49565131181975819346817833382, −5.44654856292737170112060758343, −5.37152177391065093461753563281, −5.16672051688202435136503043659, −4.57582982066390531456740611577, −4.34154644560772684971714290329, −4.18258523079743429189015388735, −4.00638054127094668878047331853, −3.60896042948492696472342169615, −3.48925901480941052106588363309, −3.39549932123022915379948786092, −2.91675076769961110135853707930, −2.65387740576460835410311137362, −2.37255062756539937161898123439, −2.10528376377791410817311469738, −2.04157745915290759940606017999, −1.04384470635856980988729038490, −0.911729691638754349029282091223, −0.804139589648096137493068581322, −0.32397315696579974703855549052,
0.32397315696579974703855549052, 0.804139589648096137493068581322, 0.911729691638754349029282091223, 1.04384470635856980988729038490, 2.04157745915290759940606017999, 2.10528376377791410817311469738, 2.37255062756539937161898123439, 2.65387740576460835410311137362, 2.91675076769961110135853707930, 3.39549932123022915379948786092, 3.48925901480941052106588363309, 3.60896042948492696472342169615, 4.00638054127094668878047331853, 4.18258523079743429189015388735, 4.34154644560772684971714290329, 4.57582982066390531456740611577, 5.16672051688202435136503043659, 5.37152177391065093461753563281, 5.44654856292737170112060758343, 5.49565131181975819346817833382, 6.03119454288081281747690953739, 6.24274651150057130257037224079, 6.35482594459534226370232423437, 6.64904589432122808647540687937, 6.67280993970042838402645348450
