| L(s) = 1 | − 110·5-s − 296·11-s + 4.44e3·19-s + 8.97e3·25-s − 540·29-s + 4.09e3·31-s + 4.79e3·41-s + 8.65e3·49-s + 3.25e4·55-s + 7.94e4·59-s − 8.45e4·61-s − 8.49e3·71-s + 7.05e4·79-s − 1.70e5·89-s − 4.88e5·95-s + 8.59e3·101-s + 7.19e4·109-s − 2.56e5·121-s − 6.43e5·125-s + 127-s + 131-s + 137-s + 139-s + 5.94e4·145-s + 149-s + 151-s − 4.50e5·155-s + ⋯ |
| L(s) = 1 | − 1.96·5-s − 0.737·11-s + 2.82·19-s + 2.87·25-s − 0.119·29-s + 0.765·31-s + 0.445·41-s + 0.514·49-s + 1.45·55-s + 2.97·59-s − 2.91·61-s − 0.200·71-s + 1.27·79-s − 2.28·89-s − 5.55·95-s + 0.0838·101-s + 0.580·109-s − 1.59·121-s − 3.68·125-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 0.234·145-s + 3.69e−6·149-s + 3.56e−6·151-s − 1.50·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.262164682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.262164682\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 22 p T + p^{5} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 8650 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 148 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 274730 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 1354590 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2220 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 11320170 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 270 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2048 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 119573530 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2398 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 288754450 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 344584890 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 827605690 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 39740 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 42298 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 1669968610 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4248 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 3239892370 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 35280 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 7103795010 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 85210 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 7720618690 T^{2} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.997237747398641560083973834876, −9.355668036513258762201909828199, −8.985464609204727567999621514017, −8.423186770586250797940167920172, −8.077400479934443942902778531018, −7.58788739672034986071780418237, −7.47366335870589045784942961799, −7.00001335816922293298812854567, −6.52255924616614531058194645684, −5.58798640295237629456379136168, −5.48695229653567532681706656070, −4.65162063057916416811697579632, −4.59151064934415568329109083736, −3.62508818867320182130947131949, −3.55316429666574274638435600366, −2.86044218469418759283312630677, −2.50743321589217937955076141410, −1.28083330032161261943830772669, −0.940046955472601381627138948055, −0.30352029737504893863687685317,
0.30352029737504893863687685317, 0.940046955472601381627138948055, 1.28083330032161261943830772669, 2.50743321589217937955076141410, 2.86044218469418759283312630677, 3.55316429666574274638435600366, 3.62508818867320182130947131949, 4.59151064934415568329109083736, 4.65162063057916416811697579632, 5.48695229653567532681706656070, 5.58798640295237629456379136168, 6.52255924616614531058194645684, 7.00001335816922293298812854567, 7.47366335870589045784942961799, 7.58788739672034986071780418237, 8.077400479934443942902778531018, 8.423186770586250797940167920172, 8.985464609204727567999621514017, 9.355668036513258762201909828199, 9.997237747398641560083973834876
