| L(s) = 1 | − 4·5-s + 52·9-s − 80·11-s + 80·19-s + 146·25-s + 280·29-s − 384·31-s − 1.04e3·41-s − 208·45-s + 932·49-s + 320·55-s + 1.39e3·59-s − 1.38e3·61-s − 1.37e3·71-s − 1.47e3·79-s + 954·81-s + 1.32e3·89-s − 320·95-s − 4.16e3·99-s − 1.41e3·101-s + 216·109-s + 1.74e3·121-s − 1.60e3·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 0.357·5-s + 1.92·9-s − 2.19·11-s + 0.965·19-s + 1.16·25-s + 1.79·29-s − 2.22·31-s − 3.99·41-s − 0.689·45-s + 2.71·49-s + 0.784·55-s + 3.07·59-s − 2.90·61-s − 2.30·71-s − 2.09·79-s + 1.30·81-s + 1.57·89-s − 0.345·95-s − 4.22·99-s − 1.39·101-s + 0.189·109-s + 1.31·121-s − 1.14·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.821841702\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.821841702\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + 4 T - 26 p T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1750 T^{4} - 52 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 932 T^{2} + 448998 T^{4} - 932 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 40 T + 1526 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 3188 T^{2} + 11156118 T^{4} - 3188 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 2756 T^{2} - 6449082 T^{4} - 2756 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 40 T + 12582 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 228 T^{2} - 31055962 T^{4} - 228 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 140 T + 47534 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 192 T + 13502 T^{2} + 192 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 127124 T^{2} + 8061820662 T^{4} - 127124 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 524 T + 203030 T^{2} + 524 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 227668 T^{2} + 24136249398 T^{4} - 227668 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 208964 T^{2} + 31759440582 T^{4} - 208964 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 29100 T^{2} + 19096738358 T^{4} + 29100 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 696 T + 346006 T^{2} - 696 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 692 T + 554862 T^{2} + 692 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 1150772 T^{2} + 511868382678 T^{4} - 1150772 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 688 T + 809582 T^{2} + 688 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 934628 T^{2} + 433634093478 T^{4} - 934628 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 736 T + 1115358 T^{2} + 736 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 846708 T^{2} + 680383628438 T^{4} - 846708 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 660 T + 1463542 T^{2} - 660 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 3172484 T^{2} + 4151279558022 T^{4} - 3172484 p^{6} T^{6} + p^{12} T^{8} \) |
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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
−10.25527865668363558814914410982, −10.21541214587252019342893328122, −9.548284382264895237578212558571, −9.350880337625508020545042412630, −8.806562427020777543442441244926, −8.688003410234071073683043577782, −8.370126726547041795488768742635, −7.87776854377764400457506383481, −7.64528526049078292612430845377, −7.42798408139642627157583569935, −6.91318418597326284111298336839, −6.81262746790073769514095299090, −6.75096687759686485093148992146, −5.65735560745928232421015020108, −5.46486368076932560364555209219, −5.46130290568458171517876492960, −4.70190385875995423008554402988, −4.54568818729749088897292605765, −4.18169280658427044106850488815, −3.43254720933684514653101032187, −3.16448488694198354765990803270, −2.67907010140260838678315132009, −1.90161720986672114510078184193, −1.39220508178131171927500657323, −0.46927494006564300260224132921,
0.46927494006564300260224132921, 1.39220508178131171927500657323, 1.90161720986672114510078184193, 2.67907010140260838678315132009, 3.16448488694198354765990803270, 3.43254720933684514653101032187, 4.18169280658427044106850488815, 4.54568818729749088897292605765, 4.70190385875995423008554402988, 5.46130290568458171517876492960, 5.46486368076932560364555209219, 5.65735560745928232421015020108, 6.75096687759686485093148992146, 6.81262746790073769514095299090, 6.91318418597326284111298336839, 7.42798408139642627157583569935, 7.64528526049078292612430845377, 7.87776854377764400457506383481, 8.370126726547041795488768742635, 8.688003410234071073683043577782, 8.806562427020777543442441244926, 9.350880337625508020545042412630, 9.548284382264895237578212558571, 10.21541214587252019342893328122, 10.25527865668363558814914410982
