L(s) = 1 | − 4·5-s − 12·11-s + 8·19-s + 11·25-s − 16·31-s + 12·41-s + 10·49-s + 48·55-s + 12·59-s + 12·61-s + 8·71-s + 16·79-s + 28·89-s − 32·95-s + 16·101-s + 4·109-s + 86·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 64·155-s + 157-s + 163-s + ⋯ |
L(s) = 1 | − 1.78·5-s − 3.61·11-s + 1.83·19-s + 11/5·25-s − 2.87·31-s + 1.87·41-s + 10/7·49-s + 6.47·55-s + 1.56·59-s + 1.53·61-s + 0.949·71-s + 1.80·79-s + 2.96·89-s − 3.28·95-s + 1.59·101-s + 0.383·109-s + 7.81·121-s − 2.14·125-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 + 5.14·155-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.099086481\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.099086481\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.786103628151456489485243511870, −8.591232557839592344483951615885, −7.915480636398502045207174142754, −7.64725977729353684673565540597, −7.57583942966493404689729995986, −7.51680130803542416449981098982, −6.92391026891074056307535428500, −6.34241770203881660642586944179, −5.53072305748612743217766137095, −5.46614414165766877410124206058, −5.03722202144207354110376420757, −4.98578568565887513645239417793, −3.96700107970389237009371843789, −3.94249437456307424426351197869, −3.24659195737275121025427609762, −3.02565526879163106159855920807, −2.34776271499390363013391814930, −2.13164458407572418486467577150, −0.70908189711121200014294892689, −0.52156910168454150213925077464,
0.52156910168454150213925077464, 0.70908189711121200014294892689, 2.13164458407572418486467577150, 2.34776271499390363013391814930, 3.02565526879163106159855920807, 3.24659195737275121025427609762, 3.94249437456307424426351197869, 3.96700107970389237009371843789, 4.98578568565887513645239417793, 5.03722202144207354110376420757, 5.46614414165766877410124206058, 5.53072305748612743217766137095, 6.34241770203881660642586944179, 6.92391026891074056307535428500, 7.51680130803542416449981098982, 7.57583942966493404689729995986, 7.64725977729353684673565540597, 7.915480636398502045207174142754, 8.591232557839592344483951615885, 8.786103628151456489485243511870