| L(s) = 1 | + 14·9-s + 116·11-s + 80·23-s − 26·25-s + 36·29-s + 436·37-s − 372·43-s + 976·53-s + 1.17e3·67-s + 2.40e3·71-s − 56·79-s − 294·81-s + 1.62e3·99-s − 184·107-s + 1.24e3·109-s − 3.91e3·113-s + 3.31e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.45e3·169-s + ⋯ |
| L(s) = 1 | + 0.518·9-s + 3.17·11-s + 0.725·23-s − 0.207·25-s + 0.230·29-s + 1.93·37-s − 1.31·43-s + 2.52·53-s + 2.14·67-s + 4.02·71-s − 0.0797·79-s − 0.403·81-s + 1.64·99-s − 0.166·107-s + 1.09·109-s − 3.26·113-s + 2.48·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 0.661·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(13.88374613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.88374613\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 14 T^{2} + 490 T^{4} - 14 p^{6} T^{6} + p^{12} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 + 26 T^{2} - 9374 T^{4} + 26 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 58 T + 3390 T^{2} - 58 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 1454 T^{2} + 214530 T^{4} - 1454 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 1376 T^{2} + p^{6} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 5246 T^{2} + 100155466 T^{4} - 5246 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 40 T + 17502 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 18 T + 43322 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 86196 T^{2} + 3514879766 T^{4} + 86196 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 218 T + 72394 T^{2} - 218 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 113360 T^{2} + 10387822434 T^{4} + 113360 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 186 T + 167550 T^{2} + 186 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 232596 T^{2} + 33146880662 T^{4} + 232596 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 488 T + 320678 T^{2} - 488 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 262606 T^{2} + 50059821354 T^{4} - 262606 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 821674 T^{2} + 270724273666 T^{4} + 821674 p^{6} T^{6} + p^{12} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 588 T + 633270 T^{2} - 588 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 1204 T + 878894 T^{2} - 1204 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 1010992 T^{2} + 489276815586 T^{4} + 1010992 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 28 T + 909886 T^{2} + 28 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 2284434 T^{2} + 1958540127914 T^{4} + 2284434 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 2263872 T^{2} + 2259994435490 T^{4} + 2263872 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 3341984 T^{2} + 4444265630850 T^{4} + 3341984 p^{6} T^{6} + p^{12} T^{8} \) |
| 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
−6.90382072081500297364115046234, −6.73673997802927068778316069583, −6.68638893680957192008339725506, −6.31618966826754834042204697594, −6.30367183955040235974279674560, −5.70839956734202483595494511300, −5.63748218954678362311180506005, −5.29730521991112322521795846649, −5.18739078324291686662018726173, −4.83981055375307676667326040484, −4.33306099585849307904428983871, −4.22708408556563881367061939228, −4.09567022426437843200910755319, −3.98347907500073684461579041634, −3.50624198381643244801100249227, −3.34183836084225552492166591592, −3.11046096880991467355778484612, −2.47668962149128583425071660293, −2.42717953890251099754563115722, −1.87713543318763576566610612310, −1.75296477952611641181123398368, −1.27180874924578794722909264041, −0.972207075156358370898291791713, −0.70471193670807707823937455503, −0.52452545637267331109800325733,
0.52452545637267331109800325733, 0.70471193670807707823937455503, 0.972207075156358370898291791713, 1.27180874924578794722909264041, 1.75296477952611641181123398368, 1.87713543318763576566610612310, 2.42717953890251099754563115722, 2.47668962149128583425071660293, 3.11046096880991467355778484612, 3.34183836084225552492166591592, 3.50624198381643244801100249227, 3.98347907500073684461579041634, 4.09567022426437843200910755319, 4.22708408556563881367061939228, 4.33306099585849307904428983871, 4.83981055375307676667326040484, 5.18739078324291686662018726173, 5.29730521991112322521795846649, 5.63748218954678362311180506005, 5.70839956734202483595494511300, 6.30367183955040235974279674560, 6.31618966826754834042204697594, 6.68638893680957192008339725506, 6.73673997802927068778316069583, 6.90382072081500297364115046234
