| L(s) = 1 | + 8·13-s + 16·17-s + 4·19-s + 8·29-s + 8·31-s − 24·37-s − 8·43-s + 4·49-s + 16·53-s − 24·59-s − 4·61-s − 8·67-s + 32·79-s − 81-s + 16·83-s − 8·97-s + 8·101-s + 32·107-s − 12·109-s + 64·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 2.21·13-s + 3.88·17-s + 0.917·19-s + 1.48·29-s + 1.43·31-s − 3.94·37-s − 1.21·43-s + 4/7·49-s + 2.19·53-s − 3.12·59-s − 0.512·61-s − 0.977·67-s + 3.60·79-s − 1/9·81-s + 1.75·83-s − 0.812·97-s + 0.796·101-s + 3.09·107-s − 1.14·109-s + 6.02·113-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.714299605\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.714299605\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 5 | $C_2^2$ | \( 1 + T^{4} \) |
| good | 7 | $C_4\times C_2$ | \( 1 - 4 T^{2} - 26 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 82 T^{4} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 136 T^{3} + 562 T^{4} - 136 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 8 T + 48 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 20 T^{3} - 146 T^{4} - 20 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 56 T^{2} + 1714 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 264 T^{3} + 2162 T^{4} - 264 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 4 T + 34 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 24 T + 288 T^{2} + 2520 T^{3} + 17426 T^{4} + 2520 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 8134 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} - 104 T^{3} - 2798 T^{4} - 104 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 24 T + 288 T^{2} + 3048 T^{3} + 27634 T^{4} + 3048 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} - 324 T^{3} - 7042 T^{4} - 324 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 88 T^{3} - 2894 T^{4} + 88 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 204 T^{2} + 19910 T^{4} - 204 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 240 T^{3} - 4174 T^{4} - 240 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 284 T^{2} + 35494 T^{4} - 284 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 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
−6.67186615815877684384997602770, −6.26907404041484912731546475499, −5.94011178039100522853940573221, −5.91766085029204398058640370245, −5.88176049230273201471352102337, −5.54701408546380266843510205623, −5.17656023002627442408976526769, −5.09791536193724352322516408638, −4.84142030830673497374182141992, −4.75033188401694827768460089236, −4.56364573441911470411031298580, −3.84598102804918472277640168603, −3.78307876764164908781937707960, −3.58382518526357906893737189225, −3.46694083812269593233847446702, −3.27821759277848823805269127287, −3.16944547317447415340715786393, −2.70018806528677082990524929180, −2.48890721805053477796537560193, −2.04830783103863816971611376061, −1.52118959870396298510793870507, −1.37194031754706712954521298490, −1.33106499070955396053600247961, −0.78192263793194354330650623638, −0.57894411095963813285998279539,
0.57894411095963813285998279539, 0.78192263793194354330650623638, 1.33106499070955396053600247961, 1.37194031754706712954521298490, 1.52118959870396298510793870507, 2.04830783103863816971611376061, 2.48890721805053477796537560193, 2.70018806528677082990524929180, 3.16944547317447415340715786393, 3.27821759277848823805269127287, 3.46694083812269593233847446702, 3.58382518526357906893737189225, 3.78307876764164908781937707960, 3.84598102804918472277640168603, 4.56364573441911470411031298580, 4.75033188401694827768460089236, 4.84142030830673497374182141992, 5.09791536193724352322516408638, 5.17656023002627442408976526769, 5.54701408546380266843510205623, 5.88176049230273201471352102337, 5.91766085029204398058640370245, 5.94011178039100522853940573221, 6.26907404041484912731546475499, 6.67186615815877684384997602770
